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Probabilistic diophantine approximation and the distribution of Halton–Kronecker sequences
Journal of Complexity 29 (2013) 397–423
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Journal of Complexity journal homepage: www.elsevier.com/locate/jco
Probabilistic diophantine approximation and the distribution of Halton–Kronecker sequences Gerhard Larcher Institute of Financial Mathematics, University of Linz, Altenberger Strasse 69, 4040 Linz, Austria
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Article history: Received 1 February 2013 Accepted 7 May 2013 Available online 21 May 2013 Keywords: Uniform distribution of sequences Discrepancy Diophantine approximation Kronecker sequences Halton sequences
abstract By a Halton–Kronecker sequence we mean a sequence in the s + t-dimensional unit-cube which is the combination of an s-dimensional Halton sequence and a t-dimensional Kronecker sequence ({n · α})n=0,1,... with α ∈ Rt . The investigation of such ‘hybrid sequences’ for their use in Monte Carlo and quasi-Monte Carlo methods first was motivated by Spanier (1995) [20]. By suitably adapting techniques of Jozsef Beck on probabilistic diophantine approximation, developed in Beck (1994) [2], we can show that for almost all α ∈ Rt for the discrepancy DN of a Halton–Kronecker (log N )s+t +ϵ
) for all ϵ > 0, which most sequence we have DN = O( N probably essentially is the best possible metrical result for this type of sequences. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Let (zn )n≥0 be a sequence in the d-dimensional unit-cube [0, 1)d . Then the discrepancy of the first N points of the sequence is defined by
AN (B) DN = sup − λ(B) , N d B⊆[0,1)
where AN (B) := #{n : 0 ≤ n < N , zn ∈ B},
λ is the d-dimensional volume and the supremum is taken over all axis-parallel subintervals B ⊆ [0, 1)d . The sequence (zn )n≥0 is called uniformly distributed if limN →∞ DN = 0.
E-mail address: [email protected]. 0885-064X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jco.2013.05.002
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G. Larcher / Journal of Complexity 29 (2013) 397–423
It is the most well-known conjecture in the theory of irregularities of distribution, that for every sequence (zn )n≥0 in [0, 1)d we have DN ≥ cd ·
(log N )d
N for a constant cd > 0 and for infinitely many N. Hence sequences whose discrepancy satisfies DN
= O
(log N )d
N
are called ‘low-discrepancy
sequences’. Note that recent investigations of Bilyk, Lacey et al., see for example [3] or [5], have led some people to conjecture that
(log N )
d+1 2
N
instead of
(log N )d N
is the best possible order for the discrepancy of sequences
in [0, 1) . At the moment the best known general lower bound for the discrepancy of sequences in [0, 1)d is d
d
DN ≥ cd ·
(log N ) 2 +ϵ(d)
N for infinitely many N, with some small ϵ(d) > 0. For more details on this topic see [1] or [4]. There are three groups of ((almost) low-discrepancy) sequences which are of main interest for applications in quasi-Monte Carlo methods. (Here by a quasi-Monte Carlo method we mean simulation in the setting of Monte Carlo methods, but using deterministic, i.e., quasi-random (usual low-discrepancy) point sets instead of pseudo-random point sets.) Indeed, these are (until now) the only known types of sequences containing concrete examples of (almost) low-discrepancy sequences. These are Kronecker sequences, Halton sequences and (T, s)-sequences in the sense of Niederreiter. The most classical type are the Kronecker sequences. A Kronecker sequence is of the form zn = ({n · α})n=0,1,... = (({n · α1 } , . . . , {n · αd }))n=0,1,... for some α = (α1 , . . . , αd ) ∈ [0, 1)d . The sequence is uniformly distributed in [0, 1)d iff 1, α1 , . . . , αd are linearly independent over Z. It was shown by Beck in his ingenious paper [2], that for almost all α in [0, 1)d and all ϵ > 0 for the discrepancy of the Kronecker sequence we have
DN = O
(log N )d+ϵ
N
.
He moreover showed that this metrical result essentially is best possible. (Indeed Beck gave an even more precise result than stated above.) So almost every Kronecker sequence is an ‘almost’ lowdiscrepancy sequence. (For d = 1 the result of Beck already was shown by Khintchine [10]. For d ≥ 2 Beck improved a result of W.M. Schmidt [19], who had given the bound DN = O
(log N )d+1+ϵ N
for almost all α .)
Halton sequences are defined as follows: Let b1 , . . . , bd ≥ 2 be pairwise coprime integers, then the Halton sequence (zn )n=0,1,... with (1)
(d)
zn = zn , . . . , zn
, in bases b1 , . . . , bd is given by
zn(j) := ψbj (n) where
ψbj (n) :=
∞
i−1 ni b − j
i =0
for n with base bj representation n=
∞
ni bij .
i=0
It is easy to show that every Halton sequence is a low-discrepancy sequence.
G. Larcher / Journal of Complexity 29 (2013) 397–423
399
The concept of (t , s)-sequences in a base b and of (T, s)-sequences in a base b (see for example [13,12,14], or [6]) was introduced by Niederreiter and it contains also earlier special examples of sequences of the same type considered by Sobol and by Faure. We do not need to explain this concept here, since we will not deal with it in this paper. We just note that it provides many concrete examples of low-discrepancy sequences. In the last years there grew considerable interest in the distribution of ‘hybrids’ of such sequences, i.e.: take an s-dimensional sequence (xn )n=0,1,... of a certain type and a t-dimensional sequence (yn )n=0,1,... of an other type and combine them to an s + t-dimensional hybrid sequence
(zn )n=0,1,... := ((xn , yn ))n=0,1,... . The use of such hybrid sequences in Monte Carlo and in quasi-Monte Carlo methods first was suggested by Spanier [20]. For the new concept of hybrid-Monte Carlo based on hybrid sequences, see for example also [17] or [7]. It is rather obvious that in general the hybrid of two low-discrepancy sequences is no longer a low-discrepancy sequence (see for example [8] where – for instance – it was shown that the hybrid of a Halton sequence in bases b1 , . . . , bd and a certain most basic low-discrepancy (t , s)-sequence in a different base b no longer is a low-discrepancy sequence). Hybrids of the two classical types of sequences, namely of Halton sequences and of Kronecker sequences – we will call them Halton–Kronecker sequences – were first studied by Niederreiter in [15,16] and in [18]. They are uniformly distributed if the Kronecker-component of the hybrid is uniformly distributed. From a metrical point of view the discrepancy of Halton–Kronecker sequences was studied in [9]. It was shown there: For every Halton sequence in [0, 1)s and almost every α ∈ Rt for the discrepancy of the s + tdimensional Halton–Kronecker sequence we have
DN = O
(log N )s+t +1+ϵ
N
(1)
for every ϵ > 0. In the case t = 1 we even have
DN = O
(log N )s+1+ϵ
(2)
N
for every ϵ > 0. To obtain these results we adapted the techniques of Khintchine [10] and W.M. Schmidt [19]. It is the aim of this paper to improve (1) to its probably (essentially) best possible form, i.e., to show Theorem 1. For every Halton sequence in [0, 1)s and almost every α ∈ Rt for the discrepancy of the s + t-dimensional Halton–Kronecker sequence we have
DN = O
(log N )s+t +ϵ
N
(3)
for every ϵ > 0. The proof of the theorem heavily depends on the deep techniques on probabilistic diophantine approximation provided by Beck in his ingenious paper [2]. Remark 1. In [2] Beck also gave a lower bound from which for example it follows that for almost all
α ∈ [0, 1)d for the discrepancy DN of the d-dimensional Kronecker sequence we have (log N )d DN = Ω . N
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G. Larcher / Journal of Complexity 29 (2013) 397–423
At the moment it seems to be out of reach to give an analogous metrical lower bound for Halton– Kronecker sequences, at least in the cases where s ≥ 2, since untilnow for s ≥ 2 it is not even known
if for the s-dimensional Halton sequence alone we have DN = Ω
(log N )s N
.
Remark 2. Until now we do not know any concrete example of a (almost) low-discrepancy Halton– Kronecker sequence. Even in the most simple case, where s = t = 1, we do not have any such concrete example. So for instance it would be of great interest to study the discrepancy DN of
√ (zn )n=0,1,... = (xn , {n · 2})
n=0,1,...
,
where xn is the one-dimensional Halton sequence in base 2, i.e., the van der Corput sequence, and to check whether
DN = O
(log N )2+ϵ
N
holds or not. The paper is organized as follows: In Section 2 we prepare the prerequisites to be able to handle the discrepancy of Halton–Kronecker sequences with the help of the Poisson-summation formula. In Section 3 we give necessary auxiliary results from probabilistic diophantine approximation, most of which are suitable generalizations of the results of Beck given in Sections 4 and 9 in [2]. In Section 4 we finish the proof of our Theorem 1. During the proof we will very often refer to the paper of Beck [2]. It is quite clear that writing our paper would not have been possible without the ingenious techniques developed by Beck in [2]. 2. Prerequisites We consider the d = s + t-dimensional sequence
(zn )n=0,1,... = ((xn , yn ))n=0,1,... , where (xn )n=0,1,... is the s-dimensional Halton sequence in pairwise coprime bases b1 , . . . , bs and (yn )n=0,1,... is the t-dimensional Kronecker sequence yn = ({n · α})n=0,1,... = (({n · α1 } , . . . , {n · αt }))n=0,1,... . Let I = [0, β) × [0, γ) ⊆ [0, 1)d with β = (β1 , . . . , βs ) and γ = (γ1 , . . . , γt ). We will choose in the following certain disjoint subsets Iint and Ibor of [0, 1)d such that Iint ⊆ I ⊆ Iint ∪ Ibor . Then with AN (I ) := #{n : 0 ≤ n < N , zn ∈ I } we obviously have
|AN (I ) − N λ(I )| ≤ |AN (Iint ) − N λ(Iint )| + AN (Ibor ) + N λ(Ibor ). We choose Iint as follows: Let
βi = (i)
with βj
β1(i) b1i
+
β2(i) b2i
+ ···
∈ {0, 1, . . . , bi − 1}.
(4)
G. Larcher / Journal of Complexity 29 (2013) 397–423
401
Then let I (j1 , . . . , js , k1 , . . . , ks , γ) :=
j −1 s i β (i) l
i=1
l=1
j i −1 βl(i) ki
+
bli
j
bii
,
l =1
+
bli
ki + 1
j
bi i
× [0, γ)
for positive integers j1 , . . . , js and ki ∈ {0, 1, . . . , bi − 1} for i = 1, . . . , s. By the construction of the Halton sequence there is a unique
j
j
r = r (j1 , . . . , js , k1 , . . . , ks ) ∈ 0, 1, . . . , b11 b22 · · · bjss − 1 such that zn ∈ I (j1 , . . . , js , k1 , . . . , ks , γ) if and only if j
n ≡ r mod (b11 · · · bjss ) and yn ∈ [0, γ) .
(5)
For x ∈ R let [x] denote the largest integer less or equal x. Then let Li := logbi N + 1 and define Iint as union of disjoint intervals by
Iint :=
L1
β
···
j 1 =1
(1)
(s)
β j −1
−1
j1 Ls
···
js =1 k1 =0
s
I (j1 , . . . , js , k1 , . . . , ks , γ).
ks =0
Further let Ibor :=
s
i =1
i−1
L Li s i βl(i) 1 βl(i) , + L × [0, 1) × [0, γ). [0, 1) × l l
j =1
l =1
bi
l =1
bi
bi i
j=i+1
Then indeed we have Iint ⊆ I ⊆ Iint ∪ Ibor , and by (4) and (5):
|AN (I ) − N λ(I )| 1 #{m | 0 ≤ n = r (j , k ) + mb(j ) < N and {nα} ∈ [0, γ)} − N ≤ λ([0, γ)) j ,k b( j ) s s 1 1 + N L +1 +N L i =1
bi i
i =1
bi i
≤ #{0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j , k ), γ + θ(j , k ))} j k s − N (j )λ ([θ(j , k ), γ + θ(j , k ))) + 1+ 3, i =1 j k where – – – – – –
# means the number of elements of a set, j = (j1 , . . . , js ), k = (k1 , . . . , ks ), θ(j , k ) := −r (j , k ) · α, intervals always are taken modulo one in each coordinate, L1 Ls j 1 =0 · · · j always means summation j s =0 ,
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–
G. Larcher / Journal of Complexity 29 (2013) 397–423
k
means summation j b11
– b(j ) := – N (j ) :=
j b22 N b(j )
· · ·
j bss ,
βj(11) −1 k1 =1
···
βj(ss) −1 k s =1
,
and
.
Since
j
1≤
s
bi (logbi N + 1),
i =1
k
we have shown that to prove Theorem 1 it suffices to show that for almost all α for all ϵ > 0 we have
|# {0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j ), γ + θ(j ))} − N (j )λ [θ(j ), γ + θ(j ))|
j
≤ c (α, ϵ, s, t , b1 , . . . , bs ) ·
(log N )s+t +ϵ
(6)
N
for all N, for all reals θ(j ) and all γ . Now it would be tempting to conclude that the assertion of Theorem 1 of our paper is a direct consequence of the above fact and of Theorem 1 in [2], since by Beck’s theorem it follows that for all j, for almost all α we have for all γ , all θ and for all ϵ > 0 the estimate
|# {0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j ), γ + θ(j ))} − N (j )λ[θ(j ), γ + θ(j ))| (log N )t +ϵ =O N
(7)
holds for all N, and that hence for almost all α we have
|# {0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j ), γ + θ(j ))} − N (j )λ [θ(j ), γ + θ(j ))|
j
=O
(log N )t +s+ϵ
N
,
(8)
with a big-O constant depending on α and on ϵ . However, of course, there is a substantial technical ‘uniformity problem’ here: Beck’s theorem guarantees that for almost every α ∈ [0, 1)d there is a constant c (α, j ) in the bigO-estimate of (7). But this constant is not necessarily bounded in j, so that argument (8) cannot be used. So we have to go through the proof of Beck and to modify and adapt his proof for our more general situation. For given j and θ(j ) = (θ1 (j ), . . . , θt (j )) we consider in analogy to [2], formula 3.11 the ‘roof-likeaverage’
¯ j (α, x; N ) := D
N (j )2
t
2
·
1
2 −
2 N (j )2 2 N (j )2
2 N (j )2
··· −
2 N (j )2
2
t
1−
−2 τ =1
+ θ(j ), x + u + θ(j ); ut +1 , N (j ) + ut +1 ) 1 − with u := (u1 , . . . , ut ), x := (x1 , . . . , xt ), and where for v = (v1 , . . . , vt ), y = (y1 , . . . , yt ) ∈ [0, 1)t
N (j )2 2
|ut +1 | 2
|uτ | Dj (b(j )α; u
du1 · · · dut dut +1 ,
(9)
G. Larcher / Journal of Complexity 29 (2013) 397–423
403
we define
D(b(j )α; v , y ; a, b) :=
1 − (b − a)
n
(yτ − vτ ),
τ =1
n
where
t
here means summation over all integers n with a < n ≤ b for which
v ≤ nb(j )α ≤ y mod [0, 1)t . Analogously to [2, 3.12.], by using Poisson’s summation formula we obtain that formally
t 1 − e2π inτ xτ
¯ j (α, x; N ) = it −1 D
n∈Zt +1 \{0}
×
1−e
τ =1
sin 2π Nn(jτ)2
2
2π Nn(jτ)2
2π i(b(j )(n1 α1 +···nt αt )−nt +1 )N (j)
2π (b(j )(n1 α1 + · · · nt αt ) − nt +1 )
×
2π nτ
sin 2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 )
2 t
2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 )
e2π iθτ (j ) .
(10)
τ =1
Here and in the following we always have n = (n1 , . . . , nt ) if n ∈ [0, 1)t , resp. n = (n1 , . . . , nt +1 ) if n ∈ [0, 1)t +1 . We set 1−e
2π inτ xτ
2π nτ
= −ixτ
and
sin 2π Nn(jτ)2
2π Nn(jτ)2
2 = 1 if nτ = 0.
The last additional factor τ =1 e2π iθτ (j ) in (10) compared with [2, 3.12] comes from the fact that we have to consider intervals [u + θ(j ), x + u + θ(j )) instead of [u, x + u) in [2]. In formulas (5.3)–(5.10) in [2] it is shown that for given j for almost , 1)t the series in t all 2απ iθ∈τ (j[0 ) (10) is absolutely convergent for all x and all N. (The additional factor τ =1 e of course does not influence this fact.) Hence for almost all α ∈ [0, 1)t the series is absolutely convergent for all j , N (j ), θ(j ) and all x, and therefore for almost all α indeed we have
t
¯ j (α, x; N ) = it −1 D
j
j
×
t 1 − e2π inτ xτ
n∈Zt +1 \{0} τ =1
2π nτ
sin 2π Nn(jτ)2 2π Nn(jτ)2
2
1 − e2π i(b(j )(n1 α1 +···nt αt )−nt +1 )N (j) 2π (b(j )(n1 α1 + · · · nt αt ) − nt +1 )
×
sin 2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 ) 2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 )
2 t
e2π iθτ (j ) .
(11)
τ =1
We will show in the following sections that for almost all α ∈ [0, 1)t for every ϵ > 0 there is a constant c (α, ϵ, s, t , b1 , . . . , bs ) such that
¯ j (α, x; N ) ≤ c (α, ϵ, s, t , b1 , . . . , bs ) D
j
holds for all N , θ(j ) and all x.
(log N )s+t +ϵ N
(12)
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G. Larcher / Journal of Complexity 29 (2013) 397–423
From (12) we derive (6) and therefore the assertion of Theorem 1 then in the following way: The left hand side of (6) is
D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )).
j
We consider α such that each coordinate of α satisfies property (13) of Lemma 1 (see Section 3). Almost all α are of this form. For these α we have for all j , θ(j ), all u = (u1 , . . . , ut ) with |uτ | ≤
2 N (j )2
for all τ = 1, . . . , t
and all ut +1 with |ut +1 | ≤ 2:
|D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )) − D(b(j )α; u + θ(j ), u + θ(j ) + γ; ut +1 , ut +1 + N (j ))| ≤ 4 + |D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )) − D(b(j )α; u + θ(j ), u + θ(j ) + γ; 0, N (j ))| t 1 ≤4+ # 0 ≤ n ≤ N (j ) {nb(j )ατ } ∈ θ (j )τ , θ (j )τ + N (j )2 τ =1 1 { } + # 0 ≤ n ≤ N (j ) nb(j )ατ ∈ θ (j )τ + γτ , θ (j )τ + γτ + N (j )2 t 1 ≤ 4 + 2t + 2 # 0 ≤ n < N (j ) ∥nb(j )ατ ∥ ≤ , N (j )2 τ =1
and hence
¯ Dj (α, x; N ) − D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )) j j t # 0 ≤ n < N (j ) ∥nb(j )ατ ∥ ≤ ≤ (4 + 2t ) + 2 j
j
τ =1
1
N (j )2
≤ c (α, ϵ, t )(log N )s+1+ϵ for every ϵ > 0 by Lemma 1. Hence our Theorem 1 follows from (12). So Sections 3 and 4 will be devoted to the proof of (12). 3. Auxiliary lemmas In this section we prove some auxiliary lemmas. Here Lemmas 3–6 essentially are adapted versions of some of the lemmas in Sections 4 and 9 in [2]. Lemma 1, which already was used in Section 2, is an easy consequence of Lemma 2. This Lemma 2 already was stated and proved in [9, Proposition 2]. Since the proof given there contained a certain gap, we use the opportunity to give a complete proof here. (Here and in the following by ‘≪’ we always mean ‘ 0 there is a c (α, ϵ) such that for all N we have
1 ≤ C (α, ϵ)(log N )s+1+ϵ . # 0 ≤ n < N (j ) ∥nb(j )α∥ < N (j )2
j
We remind that, as already defined in Section 2, by by b(j ) we mean b(j ) =
j b11
···
j bss ,
j
we mean
and by N (j ) we mean [
N b(j )
].
L1
j 1 =1
···
(13)
Ls
js =1
with Li = [logbi N ] + 1,
G. Larcher / Journal of Complexity 29 (2013) 397–423
405
As mentioned above, for the proof of Lemma 1 we need. Lemma 2. For x ∈ R let ak (x) denote the k-th continued fraction coefficient of x. For any L ∈ N let SL (α) :=
L
···
j 1 =1
L L
ak (b(j )α).
js =1 k=1
Then for almost all α ∈ R we have for every ϵ > 0 that SL (α) = O(Ls+1+ϵ ), where the implied constant is independent of L. Proof of Lemma 1. For given j let k(j ) be such that qk(j )−1 (b(j )α) < N (j ) ≤ qk(j ) (b(j )α), where qk (x) is the k-th best approximation denominator of x. Then, since
1 b(j )α − pk(j )−1 ≥ qk(j )−1 2qk(j ) · qk(j )−1 we have
# 0 ≤ n < N (j ) ∥nb(j )α∥ <
1 N (j )
≤
2
1
·
N (j )
2
1 1 2qk(j ) ·qk(j )−1
≪ ak(j ) .
So by Lemma 2
j
# 0 ≤ n < N (j ) ∥nb(j )α∥ <
for almost all α.
1 N (j )
2
≪ Slog N (α) = O (log N )s+1+ϵ
Proof of Lemma 2. For C ∈ N let EC ⊆ [0, 1) be EC := α ∈ [0, 1) | ak (b(j )α) < C (¯j1 · · · ¯js k)4 for all j1 , . . . , js ∈ N0 and all k ∈ N ,
where ¯j := max(1, j). We first show that lim λ(EC ) = 1.
C →∞
The complement E¯ C of EC contains all α for which there exist k0 and j (0) such that (0)
ak0 (b(j (0) , α)) ≥ C (¯j1 · · · ¯j(s0) k0 )4 . (0)
(0)
For such α denote by k0 (α) the minimal such k0 , by j1 (α) the minimal such j1 for the given k0 (α), (0)
(0)
(0)
further j2 (α) the minimal j2 for the given k0 (α) and j1 (α), and so on. Then E¯ C =
∞ ∞ k0 =1 (0) j 1 =0
···
∞
α ∈ E¯ C | k0 (α) = k0 , j(10) (α) = j(10) , . . . , j(s0) (α) = j(s0) .
(0) j s =0
We consider the measure of the set (0) (0) S˜ := α ∈ E¯ C | k0 (α) = k0 , j1 (α) = j1 , . . . , j(s0) (α) = j(s0) .
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G. Larcher / Journal of Complexity 29 (2013) 397–423
˜ then for the k0 − 1-st convergent If α ∈ S,
pk −1 0 qk −1 0
of b(j (0) )α we have
(0) 1 1 b(j )α − pk0 −1 < < . 2 (0) (0) 2 ¯ qk0 −1 qk0 −1 ak0 qk0 −1 (j1 · · · ¯js k0 )4 C Let α ∈ S˜ be such that this qk0 −1 = q. Then
α −
< ( 0 ) qb(j )
1
a
q2 b
(j (0) )(¯j(0) 1
· · · ¯j(s0) k0 )4 C
,
for some a ∈ {0, . . . , qb(j (0) )}, i.e., α is contained in a set of volume Since
2 (0) (0) q(¯j1 ···¯js k0 )4 C
.
(0)
q = qk0 −1 < (a1 + 1) · · · (ak0 −1 + 1) ≤ 2k0 (a1 · · · ak0 −1 ) ≤ (2C ¯j1 · · · ¯j(s0) k0 )k0 , we have (0) (0) (2C ¯j1 ···¯js k0 )k0
λ(S˜ ) < ≪
2
q=1
q(¯j1 · · · ¯js k0 )4 C
(0)
≪
(0)
log C
1
C
(¯j(10) · · · ¯j(s0) k0 )2
1
(¯j(10) · · · ¯j(s0) k0 )4 C
(0)
k0 log(C ¯j1 · · · ¯j(s0) k0 )
.
Hence λ(E¯ C ) ≪ C and the assertion limC →∞ λ(EC ) = 1 follows. Next we want to show that we have log C
ak (b(j )α)dα ≤ B(s, C ) log(kj1 · · · js ), EC
where B(s, C ) is a constant depending only on s and on C . We define ‘elementary intervals of order h’ in [0, 1) by ,...,js ) Ig(1j1,..., gh := {α ∈ [0, 1) | ai (b(j )α) = gi for all i ∈ {1, . . . , h}} .
From the fact that ψn : [0, 1) → [0, 1), x → {nx} is measure-preserving (in the sense of Lebesguemeasure) for every n ∈ N we know for all j1 , . . . , js ∈ N0 and for all h ∈ N, g1 , . . . , gh ∈ N that ,...,js ) (0,...,0) λ(Ig(1j1,..., gh ) = λ(Ig1 ,...,gh ),
and together with [11, Eq. (57)] we derive for all j1 , . . . , js ∈ N0 that ,...,js ) λ(Ig(1j1,..., gh−1 ,gh ) =
2 gt2
,...,js ) λ(Ig(1j1,..., gh−1 ).
Hence, and by the definition of EC we have
ak (b(j )α)dα ≤ EC
=
C (¯j1 ···¯js k)4
∞
gh =1
g1 =1
C (¯j1 ···¯js k)4
∞
gh =1
g1 =1
C (¯j1 ···¯js k)4
=
gh =1 C (j1 ···js k)4
=
gh =1
···
gh−1 =1
···
∞ 2
gh 2 gh
∞
g1 =1
∞
(j ,...,js ) Ig 1,...,g 1
ak (b(j )α)dα
h
,...,js ) gh λ(Ig(1j1,..., gh−1 ,gh )
gh−1 =1
···
∞
,...,js ) λ(Ig(1j1,..., gh−1 ,gh )
gh−1 =1
≤ B(s, C ) log(¯j1 · · · ¯js k).
G. Larcher / Journal of Complexity 29 (2013) 397–423
407
So we have
L
L L
···
EC j =1 1
j s =1 k =1
ak (b(j )α)
ϵ 1+ s+ 1
(¯j1 · · · ¯js k)
dα ≤
L
···
j1 =1
L L B(C , s) log(k¯j1 · · · ¯js ) ϵ
js =1 k=1
(¯j1 · · · ¯js k)1+ s+1
dα
≪ const (C , ϵ). Since the integrand above is monotonically increasing in L, we have that for almost all α ∈ EC it is L
···
j1 =1
L L
ak (b(j )α)
js =1 k=1
(¯j1 · · · ¯js k)1+ s+1
ϵ
dα ≪ const (C , ϵ)
and hence SL (α) ≪ const (C , ϵ)Ls+1+ϵ for all L ∈ N.
The next lemma is the analogue to Lemma 4.4 in [2]. Lemma 3. For almost all α ∈ [0, 1)t for all ϵ > 0 there is a c (α, ϵ) such that
j
(∥nb(j )α∥)
t
−1
n¯ τ
¯ |n1 · · · nt |)t +2 (log
≤ c (α, ϵ)(log N )s+ϵ
τ =1
n∈Zt \{0}
for all N. ¯ := max(1, log |x|). Here and in all the following n¯ τ := max(1, |nτ |) and logx Proof of Lemma 3. By Lemma 3 in [9] we know that for almost all α ∈ [0, 1)t there is a c (α) > 0 such that
∥nb(j )α∥ > c (α)(|¯n1 · · · n¯ t | · (¯j1 · · · ¯js ))−3 for all j and all n ∈ Zt \ {0}, i.e., 1
|log ∥nb(j )α∥ |
≫
c (α) . ¯ log(|n1 · · · nt | · (¯j1 · · · ¯js ))
(14)
In analogy to Schmidt [19] we consider the integral
Jδ (j , n) :=
[0,1)t
−1 ∥nb(j )α∥ · |log ∥nb(j )α∥ |1+δ dα,
which has a finite value c (δ) for any δ > 0, that is independent of j and n. Hence with δ := t +1 1 and for every ϵ > 0 we have
[0,1)t
−1 ∥nb(j )α∥ · |log ∥nb(j )α∥ |1+δ j
n∈Zt \{0}
×
t
−1 1+ ϵ ϵ 1+ 2s 2s ) · ¯j1 · · · ¯js dα
¯ τ) (¯nτ (logn
1+δ
τ =1
=
j
n∈Zt \{0}
Jδ (j , n)
t
τ =1
−1 1+ ϵ ϵ 1+ 2s 1+δ 2s ¯ ¯ ¯ (¯nτ (log nτ ) ) · j1 · · · js ≪ c (ϵ),
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G. Larcher / Journal of Complexity 29 (2013) 397–423
for all N, and therefore for almost all α there is a c (α, ϵ) such that
−1 ∥nb(j )α∥ · |log ∥nb(j )α∥ |1+δ n∈Zt \{0}
j
×
t
−1 1+ ϵ ϵ 1+ 2s 2s · · · ¯js < c (α, ϵ) ) · ¯j1
¯ nτ ) (¯nτ (log
1+δ
τ =1
for all N. By (14) therefore for almost all α there is a c˜ (α, ϵ) such that
∥nb(j )α∥ ·
ϵ ϵ 1+ 2s 1+ 2s 1+δ ¯ ¯ ¯ (¯nτ (log nτ ) ) · j1 · · · js
τ =1
n∈Zt \{0}
j
t
−1 ¯ (|n1 · · · nt | · (¯j1 · · · ¯js ))) × (log
≤ c˜ (α, ϵ),
1+δ
for all N. Hence, using that
¯ |n1 · · · nt | · log ¯j1 · · · log ¯js , ¯ (|n1 · · · nt | · (¯j1 · · · ¯js ))1+δ ≪ log log for almost all α there is a c¯ (α, ϵ) such that
j
∥nb(j )α∥ ·
t
(t +1)(δ+1)
¯ |n1 · · · nt |) (log
n¯ τ
τ =1
n∈Zt \{0}
s
1+ ϵ ji s
−1
¯
≤ c¯ (α, ϵ)
i =1
for all N, and therefore, for almost all α and all ϵ > 0 we have
j
∥nb(j )α∥ ·
−1
n¯ τ
(t +2)
¯ |n1 · · · nt |) (log
≪ c¯ (α, ϵ)(log N )s+ϵ
τ =1
n∈Zt \{0}
for all N.
t
The following Lemma 4 is the generalization of Lemma 4.1 in [2], however in a slightly weaker form. Lemma 4. For almost all α ∈ Rt for every ϵ > 0 there is a c (α, ϵ) such that
j
∥nb(j )α∥ ·
n∈Zt \{0} C11 ,C12
t
−1 n¯ τ
≤ c (α, ϵ)(log N )s+t +ϵ
τ =1
for all N. Here the conditions C11 and C12 mean C11 : nτ ≤ N 3
for τ = 1, . . . , t .
C12 :
t
n¯ τ
· ∥nb(j )α∥ < (log N )20(s+t ) .
τ =1
Proof of Lemma 4. For the proof of Lemma 4 we will need Lemma 5.
G. Larcher / Journal of Complexity 29 (2013) 397–423
409
Let α ∈ Rt be one of the almost all α satisfying Lemma 5(a) and (b), let ϵ > 0 and c˜ (α, ϵ) like in Lemma 5(b) (note there is a second constant c¯ different from c˜ in the expression below), then:
j
∥nb(j )α∥ ·
t
−1 n¯ τ
τ =1
n∈Zt \{0} C11 ,C12
l¯2 · · · l¯2t 1
c¯ log log N
≤
l ∈ Nt 0 l 3 3 τ ≤N 3 τ =1,...,t
j
v=−˜c (α,ϵ) log log N n∈l-box
2v−1
3l1 +···lt . (15)
C13
Here we have to explain what we mean, by an l-box, by condition C13 in (15) by c¯ , and why we can choose the bounds for v in this way: For given l = (l1 , . . . , lt ) ∈ Nt0 by the l-box we mean the set
lτ −1
n ∈ Zt | 33
lτ ≤ n¯ τ < 33 for τ = 1, . . . , t ,
lτ −1
where we set 33 = 1 if lτ = 0, and ¯l := max(1, l). By condition C13 we mean C13 :
2v−1
¯ · · ·¯
3l1 +···+lt l21
l2t
≤ ∥nb(j )α∥ ·
t
n¯ τ
2v
<
¯ · · · ¯l2t
3l1 +···+lt l21
τ =1
.
Note that 2v
¯ · · ·¯
3l1 +···+lt l21
l2t
≪
2v
¯ n¯ 1 · · · log ¯ n¯ t · ¯l21 · · · ¯l2t log
therefore by Lemma 5(b) the inner sum
n∈l-box C13
,
would be an empty sum for v < −˜c (α, ϵ) log log N.
Finally c¯ is determined such that 2c¯ log log N 3l1 +···+lt ¯l21 · · · ¯l2t
> (log N )20(s+t ) .
This is possible since 2c¯ log log N 3l1 +···+lt ¯l21 · · · ¯l2t
≫
2c¯ log log N
(log N )t (log log N )2t
> (log N )20(s+t )
for c¯ large enough in dependence only on s and t. Hence the inner sum
empty sum for v > c¯ log log N. We proceed with the estimation of the right hand side of (15) which is c¯ log log N
≤
l ∈Nt 0 l 33 τ ≤N 3 τ =1,...,t
¯l21 · · · ¯l2t
v=−˜c (α,ϵ) log log N
2v−1
3l1 +···lt
j
1,
n∈l-box C13
and by Lemma 5(a) – with c (α, ϵ) like in this Lemma 5(a) – this is at most c¯ log log N
l ∈Nt 0 l 3 3 τ ≤N 3 τ =1,...,t
v=−˜c (α,ϵ) log log N
¯l21 · · · ¯l2t 2v−1
3l1 +···lt · c α,
ϵ
ϵ 2
ϵ
· 2v c˜ (α, ϵ) log log N max(1, v)1+ 2 (log N )s+ 2
n∈l-box C13
in (15) would be the
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G. Larcher / Journal of Complexity 29 (2013) 397–423
ϵ ϵ ϵ ¯l21 · · · ¯l2t 3l1 +···+lt (log N )s+ 2 (log log N )3+ 2 · ≪ c ′ α, 2
l ∈ Nt 0 l 3 3 τ ≤N 3 τ =1,...,t
≪ c ′′ (α, ϵ)(log N )s+t +ϵ . Lemma 5. (a) For almost all α for all ϵ > 0 there is a c (α, ϵ) such that for all V ∈ N, all v ≥ −V , all N and all l ∈ Nt0 we have
2v # n ∈ l-box ∥nb(j )α∥ < 2 2 ¯ n¯ 1 · · · logn ¯ t ¯l1 · · · ¯lt n¯ 1 · · · n¯ t log
j
≤ c (α, ϵ) · 2v V max(1, v)1+ϵ (log N )s+ϵ . (b) For almost all α for all ϵ > 0 there is a c˜ (α, ϵ) such that for all N and all v < −˜c (α, ϵ) log log N, all l ∈ Nt0 and for all j we have
2v # n ∈ l-box ∥nb(j )α∥ < 2 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯lt n¯ 1 · · · n¯ t logn
= 0.
Proof of Lemma 5. (a) For ϵ > 0, V > 0 and v ≥ −V consider v
[0,1)t w=−V
1 2w V
max(1, w)
1+ϵ
(¯j1 · · · ¯js )−1−ϵ l ∈Nt0
j
2w dα × # n ∈ l-box ∥nb(j )α∥ < 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯l2t n¯ 1 · · · n¯ t logn v 1 −1−ϵ ¯ ¯ = (j1 · · · js ) 1dα, 2w V max(1, w)1+ϵ j t n∈l-box A w=−V
l ∈N0
where A :=
α ∈ Rt ∥nb(j )α∥ <
2w
¯ 1 · · · logn ¯ t ¯l21 · · · ¯l2t n¯ 1 · · · n¯ t logn
.
Since
1dα = A
2w
¯ 1 · · · logn ¯ t ¯l21 · · · ¯l2t n¯ 1 · · · n¯ t logn
the last expression is
≤
v
1 V max(1, w)
1+ϵ
w=−V
(¯j1 · · · ¯js )−1−ϵ j
l ∈Nt0
1
¯l21 · · · ¯l2t
1
n∈l-box
¯ 1 · · · logn ¯ t n¯ 1 · · · n¯ t logn
≪ c (ϵ).
¯ 1 · · · logn ¯ t )−1 ≪ 1.) (Note that n∈l-box (¯n1 · · · n¯ t logn Therefore for almost all α for all ϵ > 0 there is a c (α, ϵ) such that for all v ≥ −V and all N we have
l ∈Nt0
j
2v # n ∈ l-box ∥nb(j )α∥ < 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯l2t n¯ 1 · · · n¯ t logn (¯j1 · · · ¯js )1+ϵ 1
≤ c (α, ϵ)2v V max(1, v)1+ϵ (log N )s+ϵ . From this part (a) of Lemma 5 immediately follows.
G. Larcher / Journal of Complexity 29 (2013) 397–423
411
(b) Let α be such that part (a) of Lemma 5 is satisfied, and let c (α, ϵ) be the constant from part (a) of Lemma 5 and c˜ (α, ϵ) be such that c (α, ϵ)2−V V (log N )s+ϵ < 1 for V > c˜ (α, ϵ) log log N. Then for all such V and v = −V by part (a) we have
2v =0 # n ∈ l-box ∥nb(j )α∥ < 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯l2t n¯ 1 · · · n¯ t logn
for all l and j, and part (b) follows.
Lemma 6, finally, is a generalization of Lemma 4.2 in [2]. Lemma 6. For j given, let Vj = (vj (1), . . . , vj (t )), Wj = (wj (1), . . . , wj (t )) be such that 0 ≤ vj (τ ) < wj (τ ) for τ = 1, . . . , t. For n = (n1 , . . . , nt ) ∈ Nt0 we write Vj ≤ n < Wj if vj (τ ) ≤ nτ < wj (τ ) for τ = 1, . . . , t. For arbitrary cj = q4j with some qj ∈ N let
Z (α, j , cj , Vj , Wj ) := # n Vj ≤ n < Wj with {nb(j )α } <
cj
n¯ 1 · · · n¯ t
and cj
E (j , cj , Vj , Wj ) :=
Vj ≤n
n¯ 1 · · · n¯ t
.
Let J ⊆ {j = (j1 , . . . , jt ) | 0 ≤ ji < Li for i = 1, . . . , s} . Then for almost all α for all ϵ > 0 there is a c (α, ϵ) such that for all J , Vj , Wj and qj we have
Z (α, j , cj , Vj , Wj ) =
j ∈J
E (j , cj , Vj , Wj ) + E
j ∈J
with
|E | ≤ c (α, ϵ)
3
1
ϵ
cj4 (log cj )t (¯j1 · · · ¯js ) 2 + 2 (log N )
1 +ϵ t 2
.
j ∈J
Proof of Lemma 6. We proceed quite analogously (with adapted details) as Beck in the proof of Lemma 4.2 in [2]. For some parts of the proof of our Lemma 6 we will directly refer to the proof of Lemma 4.2. in [2]. So we will also refer for some of the necessary definitions (which are used in this proof only) to the proof of 4.2 in [2], and we do not give them all here in this paper again. By Lemma 9.3. in [2] for every j we have
[0,1)t
(Z (α, j , cj , Vj , Wj ) − E (j , cj , Vj , Wj ))2 dα = 4
Vj ≤n
div(n)
cj n¯ 1 · · · n¯ t
,
where div(n) = div(n1 , . . . , nt ) denotes the number of common divisors of n1 , . . . , nt . In the same way as in the proof of Lemma 4.2 in [2, pp. 496 and 497], from this we obtain (for the definitions of ‘special boxes’, ‘size’, ‘order’ see there):
C14
[0,1)t
˜ ) − E (j , cj , V˜ , W ˜ ))2 dα ≪ htj · cj · (2p )t . (Z (α, j , cj , V˜ , W
(16)
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G. Larcher / Journal of Complexity 29 (2013) 397–423
Here condition C14 means C14 : ˜ ) in the sense of Lemma 4.2 in [2] of size
p
The integer p is defined such that e2 ≤ N < e2 . For κ > 0 and ϵ > 0 let (note that cj = q4j for some qj = 1, 2, . . .)
˜ ) − E (j , cj , V˜ , W ˜ ))2 Bκ (p, hj , j , qj ) := α (Z (α, j , cj , V˜ , W C 14 ≥ κ · htj · q6j · (2p )t · p2 · (¯j1 · · · ¯js )1+ϵ . Then for the measure of Bκ from (16) it follows that
λ(Bκ (p, hj , j , qj )) ≪
1
κ · q2j · p2 · (¯j1 · · · ¯js )1+ϵ
and therefore
λ(Bκ (p, hj , j , qj )) ≤
j ,p,qj
1
κ
c (ϵ).
With Bκ :=
j ∈J p,qj
Bκ (p, hj , j , qj ) c (ϵ)
then we have λ(Bκ ) ≤ δ for κ ≥ δ and for every J. Let now hj := pt + c¯ log q4j for a certain absolute constant c¯ such that q4i · 2tp −hj ≪ 1. Let ϵ > 0, then for almost all α ∈ Rt there is a κ(ϵ, t ) such that α ̸∈ Bκ for all κ > κ(ϵ, t ). That means, for almost all α ∈ Rt there is a c (α, ϵ, t ) such that for all j , p, qj (with hj chosen like above) we have
˜ ) − E (j , cj , V˜ , W ˜ ))2 ≪ c (α, ϵ, t ) · htj · q6j · (2p )t · p2 · (¯j1 · · · ¯js )1+ϵ . (Z (α, j , cj , V˜ , W C14
From this, exactly in the same way as it is carried out in the proof of Lemma 4.2. in [2] on pp. 498 and 499 we obtain: (note that there are typing errors in formulas (9.66) and (9.67) in [2], the correct form of (9.66) is
2
˜ ) − E (c ; V˜ , W ˜ ) ≪ · · · Z (α; c , V˜ , W
˜) some special boxes Q (V˜ ,W
and the correct form of (9.67) is
Z α; c , upper lower
bound
−E
upper lower
≪ · · · .) bound
For α and ϵ > 0 from above, we have for all j , p, qj that
1 ϵ t ˜ ˜ ˜ ˜ Z (α, j , cj , V , W ) − E (j , cj , V , W ) ≪ c (α, ϵ, t ) · htj · q3j · (2p ) 2 · p · (¯j1 · · · ¯js ) 2 + 2 . C 14
G. Larcher / Journal of Complexity 29 (2013) 397–423
413
Hence (again using the arguments of Beck on pp. 498 and 499) for almost all α
Z (α, j , cj , Vj , Wj ) =
j ∈J
E (j , cj , Vj , Wj ) + E
j ∈J
with (note that hj is chosen such that q4j · 2tp−hj ≪ 1 and that p ≪ log log N)
|E | ≪
1
t
ϵ
c (α, ϵ, t ) · htj · q3j · (2p ) 2 · p · (¯j1 · · · ¯js ) 2 + 2 + q4j · 2tp−hj
j ∈J
≪ c (α, ϵ)
3
1
ϵ
cj4 (log cj )t (¯j1 · · · ¯js ) 2 + 2 (log N )
1 +ϵ t 2
j ∈J
for all N , J , Vj , Wj , qj = 1, 2, . . . and cj = q4j .
4. The proof of the theorem For the proof of the theorem we can restrict to t ≥ 2, since it already was shown for t = 1 in [9]. t By the considerations in Section 2 it remains to show that for almost all α ∈ R we have that the (for almost all α) absolutely convergent series satisfies for all ϵ > 0:
:=
j
n∈Zt +1 \{0}
pro(N , x, θ, j , n) ≤ c (α, ϵ, b1 , . . . , bs )
(log N )s+t +ϵ N
for arbitrary x = (x1 , . . . , xt ) and arbitrary θ = θ(j ) = (θ1 (j ), . . . , θt (j )), where pro(N , x, θ, j , n) := i
t −1
t 1 − e2π inτ xτ 2π nτ
τ =1
· g (n, j ) ·
t
1 − e2π i(b(j )(n1 α1 +···+nτ ατ )−nt +1 )N (j ) 2π (b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )
e2π iθτ (j ) ,
τ =1
where
2 2 t sin 2π Nn(jτ)2 sin(2π (b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )) g (n, j ) := · . 2π Nn(jτ)2 2π (b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 ) τ =1 2π inτ xτ = −ixτ and Here we always – without extra noting it – assume 1−e2π n τ
nτ N (j )2 nτ N (j )2
sin 2π 2π
= 1 if nτ = 0.
Note that always
|pro(N , x, θ, j , n)| 2 −1 nτ · max 1, ≤ · n¯ 1 · · · n¯ t |(b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )| N (j )2 τ =1 1
≤
1
1 n¯ 1 · · · n¯ t
·
1
|(b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )|
≤
t
1
1
n¯ 1 · · · n¯ t ∥nb(j )α∥
holds. If in the following we write LHS ≪ RHS, then we always mean that LHS ≤ c · RHS, with a constant c depending at most on t and on s (and not on N , x, α, . . .). We now proceed essentially along the same lines as Beck did in [2], by dividing in various parts, but with the necessary adaptations. Let first
1
:=
j
n∈Zt \{0} nt +1 ∈Z C1
pro(N ; x, θ, j , n),
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G. Larcher / Journal of Complexity 29 (2013) 397–423
where condition C1 means: C1 : n = (n1 , . . . , nt ) with max |nτ | > N (j )2 (log N (j ))t . 1≤τ ≤t
Then
≪
1
2
+
, 3
where
2 −1 nτ , · max 1, N (j )2 τ =1
:=
j
2
(|¯n1 · · · n¯ t | · ∥nb(j )α∥)
−1
n∈Zt \{0} C1
t
and
:=
j
3
∞
t
τ =1
r =1 n∈Zt \{0}
2 −1 nτ 1 |nτ | max 1, · 3. 2 N (j ) r
C1
Now (to obtain the following first inequality set max |nτ | := κ · N (j )2 · (log N (j ))t with κ ≥ 1 in the above expression and the result will follow)
≪
max(1, |nτ |(log |nτ |)2 )
j
3
≪
2
n∈Zt \{0} τ =1
∞ j
For
∞ −1 1
t
n =2
r =1
t
1 n(log n)2
·
∞ 1 r =1
r3
r3
≪ (log N )s .
we obtain in the same way as above (note that we can restrict to t ≥ 2):
≪
j
2
τ =1
n∈Zt \{0}
≪
j
−1 t 2t ¯ 1 · · · nt ) ∥nb(j )α∥ max(1, logn n¯ τ
t ¯ 1 · · · nt ) t + 2 ∥nb(j )α∥ max(1, logn n¯ τ
−1 ,
τ =1
n∈Zt \{0}
hence by Lemma 3 it follows, that for almost all α for every ϵ > 0 there is a c (α, ϵ) such that
≪ c (α, ϵ)(log N )s+ϵ .
2
Next we handle
4
:=
j
pro(N ; x, θ, j , n),
n∈Zt \{0} C2 ,C3
where condition C2 means: C2 : n = (n1 , . . . , nt ) with max |nτ | ≤ N (j )2 (log N (j ))t 1≤τ ≤t
and where condition C3 means: C3 :
|b(j )(n1 α1 + · · · + nt αt ) − nt +1 | ≥
1 3
.
G. Larcher / Journal of Complexity 29 (2013) 397–423
415
The inner sum n in 4 is estimated for every α in exactly the same way as the sum 4 in [2, Section 5], to be ≪ (log N )t (notethat in [2] the letter k is our parameter t), with an implied constant independent of j. Hence our 4 satisfies
≪ (log N )s+t
4
for every α. Next let
:=
j
5
pro(N ; x, θ, j , n),
n∈Zt \{0} nt +1 ∈Z C5 C2 ,C4
where condition C4 means: C4 :
|b(j )(n1 α1 + · · · + nt αt ) − nt +1 | ≤
(log N )20(t +s) |¯n1 · · · n¯ t |
and where condition C5 means: C5 : nt +1 is such that |b(j )(n1 α1 + · · · nt αt ) − nt +1 | = ∥nb(j )α∥ . Then by Lemma 4 for almost all α there is for all ϵ > 0 a c (α, ϵ) such that
≤
j
5
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1 ≤ c (α, ϵ)(log N )s+t +ϵ .
n∈Zt \{0} C2 , C4
In the next step we handle two parts of the following sum
:=
j
6
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1
n∈Zt \{0} C6 ,C7
where condition C6 means: C6 :
|nτ | ≤ N (j )2 (log N (j ))t for τ = 1, . . . , t , and where condition C7 means: C7 : 1 3
> ∥nb(j )α∥ >
The two parts
(j )
71
(log N )20(t +s) . |¯n1 · · · n¯ t |
and (j )
72
of
6
which we treat first are the following:
let vj = (v1 , . . . , vt ) with (j ) 1 ≤ 2vτ ≤ N (j )2 (log N (j ))t
for all τ = 1, . . . , t ,
and (j ) (j ) T (vj ) := n ∈ Zt \ {0} | 2vτ −1 ≤ n¯ τ < 2vτ , 1 ≤ τ ≤ t ,
then
71
:=
j
vj
n∈T (vj )
(j ) min vτ ≪log log N τ
C6 ,C7
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1
416
G. Larcher / Journal of Complexity 29 (2013) 397–423
and
:=
72
j
vj
n∈T (vj )
(j ) N (j )2 max 2vτ ≥ 4 τ
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1 .
C6 ,C7
Let us consider the first part:
≪
71
j
vj
1
(j ) min vτ ≪log log N τ
q(j ) C8
2q(j )
#T (vj , q(j ), j )
where condition C8 means: C8 :
(log N )20(t +s) ≤ 2q(j ) ≤ 2N 2t (log N )20(t +s)t , and where T (vj , q(j ), j ) := n ∈ T (vj )|2q(j )−1 < n¯ 1 · · · n¯ t ∥nb(j )α∥ ≤ 2q(j ) .
(Note, that n¯ 1 · · · n¯ t ∥nb(j )α∥ ≤ N (j )2t (log N (j ))t
2
always trivially holds, since nτ < N (j )2 (log N (j ))t for all τ .) Hence
71
1
≪
v =(v1 ,...,vt ) min vτ ≪log log N
τ
q C8
2q
#T (v , q, j ).
j
By Lemma 6, for almost all α for all ϵ > 0 there is a c (α, ϵ) such that
# T (v , q, j ) ≪
j
2q−1
j
(j) (j) 2vτ −1 ≤¯nτ <2vτ
+ c (α, ϵ)
n¯ 1 · · · n¯ t
3q 1 ϵ 2 4 +ϵ (¯j1 · · · ¯js ) 2 + 2 (log N )
j
≪ c (α, ϵ)2q (log N )s since 2q ≥ (log N )20(t +s) . Hence
71
≪ c (α, ϵ)
v =(v1 ,...,vt ) min vτ ≪log log N
τ
(log N )s q C8
≪ c (α, ϵ)(log N )t −1 log log N · log N · (log N )s ≪ c (α, ϵ)(log N )t +s log log N . The second part is handled quite analogously and leads to
72
≪ c (α, ϵ) ·
v =(v1 ,...,vt ) 2 N (j)2 N (j)2 (log N (j))t >max 2vτ > 4 τ
(log N )s q C8
≪ c (α, ϵ)(log N )t −1 log log N · log N · (log N )s ≪ c (α, ϵ)(log N )t +s log log N .
1 +ϵ t 2
G. Larcher / Journal of Complexity 29 (2013) 397–423
417
So it remains to estimate the following sum (i.e., the analogue to the work done by Beck in his Sections 6 and 7 in [2]):
:=
pro(N ; x, θ, j , n),
n∈Zt \{0} C7 ,C9
j
8
where condition C9 means: C9 :
(log N )20(t +s) ≤ min |nτ | ≤ max |nτ | ≤
N (j )2 4
.
(Note that condition C7 implies C5 .) It is not sufficient to handle 8 by just estimating absolute values of the summands. So in analogy to Beck [2, p. 469], we write (here and in the following an nτ in the denominator which equals 0 is replaced by 1)
=
8
i t −1
(2π )t +1
Λ(θ, j )
n∈Zt \{0} C7 ,C9
j
n1 · · · nt (nb(j )α − nt −1 )
δ1 +···+δt +1
×
g (n, j )
Λ(θ, j )
j
e2π iL1,j (n,α)
(−1)
n1 · · · nt (nb(j )α − nt +1 )
n∈Zt \{0} C7 , C9
1=(δ1 ,...,δt +1 )∈{0,1}t +1
+
where t
Λ(θ, j ) :=
e2π iθτ (j ) ,
τ =1
and L1,j (n, α) := δt +1 N (j )(nb(j )α − nt +1 ) +
t
δτ xτ nτ .
τ =1
We start the analysis with the first part of
:=
9
Λ(θ, j )
8,
i.e., with
g (n, j )
n∈Zt \{0} C7 ,C9
j
n1 · · · nt (nb(j )α − nt −1 )
.
For 1
, (log N )2 l = (l1 , . . . , lt +1 ) ∈ Nt0+1 δ :=
and
ϵ = (ϵ1 , . . . , ϵt +1 ) ∈ {−1, 1}t +1 let
U (l , ϵ, j ) := n ∈ Zt | (1 + δ)lτ ≤ ϵτ nτ < (1 + δ)lτ +1
for 1 ≤ τ < t ,
and
(1 + δ)lt +1 ≤ ϵτ +1 n1 · · · nτ (nb(j )α − nt +1 ) < (1 + δ)lt +1 +1 .
g (n, j ) ,
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G. Larcher / Journal of Complexity 29 (2013) 397–423
By C9 for given j the number of sets U (l , ϵ, j ) which are relevant for
log N t +1
8
(and therefore for
. at most ≪ δ For any choice of
ϵ˜ := (ϵ1 , . . . , ϵt ) ∈ {−1, 1}t let
ϵ˜ + := (ϵ1 , . . . , ϵt , 1) ∈ {−1, 1}t +1 and
ϵ˜ − := (ϵ1 , . . . , ϵt , −1) ∈ {−1, 1}t +1 . Then
=
Λ(θ, j )
j
9
l
ϵ˜ ∈{−1,1}t
n∈U (l ,˜ϵ+ ,j )
n1 · · · nt (nb(j )α − nt −1 )
g (n, j )
+
g ( n, j )
n∈U (l ,˜ϵ− ,j )
n1 · · · nt (nb(j )α − nt −1 )
=
l ,˜ϵ
Λ(θ, j )
+
n∈U (l ,˜ϵ+ ,j )
j
.
n∈U (l ,˜ϵ− ,j )
We will need now the following technical lemma
˜ ∈ U (l , ϵ˜ , j ) satisfying C7 and C9 , then Lemma 7. Let n ∈ U (l , ϵ˜ , j ) satisfying C7 and C9 and n +
−
˜ , j) g (n, j ) δ g (n + n · · · n (nb(j )α − n ) n˜ · · · n˜ (nb(j )α − n˜ ) ≪ (1 + δ)lt +1 . 1 t t +1 1 t t +1 Proof of Lemma 7. Note that 0 < nb(j )α − nt +1 <
1 3
and
−
1 3
˜ (j )α − n˜ t +1 < 0. < nb
Further 1 3
> |nb(j )α − nt +1 | ≥
(1 + δ)lt +1 = (1 + δ)lt +1 −(l1 +···lt )−t (1 + δ) · · · (1 + δ)lt +1 l1 +1
hence lt +1 ≤ (l1 + · · · + lt ) + t . We want to show
˜ , j) g (n, j ) g (n δ ≪ − . n1 · · · nt |nb(j )α − nt +1 | n˜ 1 · · · n˜ t nb(j )α − n˜ t +1 (1 + δ)lt +1 If we can show, that
g (n, j ) − g (n˜ , j ) ≪ δ,
9)
is
G. Larcher / Journal of Complexity 29 (2013) 397–423
419
then we have
˜ , j) + ϵ g (n, j ) = g (n for some ϵ with |ϵ| ≪ δ , hence:
˜ , j) g (n g (n, j ) − n1 · · · nt |nb(j )α − nt +1 | n˜ 1 · · · n˜ t nb ˜ (j )α − n˜ t +1 ˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb 1 (g (n˜ , j ) + ϵ) − g (n˜ , j ) , = n1 · · · nt |nb(j )α − nt +1 | ˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb and
˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb − g (n˜ , j ) (g (n˜ , j ) + ϵ) n1 · · · nt |nb(j )α − nt +1 | ˜ (j )α − n˜ t +1 ˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb n˜ 1 · · · n˜ t nb = g (n˜ , j ) −1 +ϵ n1 · · · nt |nb(j )α − nt +1 | n1 · · · nt |nb(j )α − nt +1 | 1 ≤ max ((1 + δ) − 1) , 1 − + δ(1 + δ) ≪ δ, (1 + δ) hence
˜ , j) δ g (n, j ) g (n ≪ − . n1 · · · nt |nb(j )α − nt +1 | n˜ 1 · · · n˜ t nb(j )α − n˜ t +1 (1 + δ)lt +1 It remains to show that
g (n, j ) − g (n˜ , j ) ≪ δ. Note that N2π2 (njτ) ≤ π2 and |2π (nb(j )α − nt +1 )| ≤ Using show
sin x x
= 1−
x2
+ ϵ(x)x with |ϵ(x)| ≤ 4
6
2π 3
1 120
.
for − 23π ≤ x < 23π , it now is an easy exercise to
g (n, j ) − g (n˜ , j ) ≪ (1 + δ)lt +1 −l1 −···−lt δ ≪ δ, since lt +1 ≤ (l1 + · · · + lt ) + t . The proof of Lemma 7 is finished.
We proceed now with the proof of the theorem, i.e., first with the estimation of For l , ϵ˜ given, we consider
˜ := Λ(θ, j )
n∈U (l ,˜ϵ+ ,j )
j
+
n∈U (l ,˜ϵ− ,j )
Let U :=
j
g (n, j )
U (l , j , ϵ˜ ). +
n1 · · · nt (nb(j )α − nt −1 )
g (n, j ) n1 · · · nt (nb(j )α − nt −1 )
.
9:
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G. Larcher / Journal of Complexity 29 (2013) 397–423
By Lemma 6 we have for almost all α for all ϵ > 0, with C1 := (1 + δ)lt +1 +1 and C2 := (1 + δ)lt +1 : #U =
C1 − C2
n∈Zt \{0}
j
+ R ≤ (C1 − C2 )δ t (log N )s + R
n¯ 1 · · · n¯ t
(1+δ)lτ ≤nτ <(1+δ)lτ +1
with 3 1 1 +ϵ t 4 +ϵ + 2ϵ 2 ¯ ¯ 2 |R| ≪ c (α, ϵ) (j1 · · · js ) (log N ) C1 . j
g (n,j )
Hence (note that |Λ(θ, j )| ≤ 1 and n ···n (nb(j )α−n ) ≤ t 1 t −1
1 ), C1
altogether we have:
˜ ≤ (C − C )δ t (log N )s δ + c (α, ϵ) (¯j · · · ¯j ) 12 + 2ϵ (log N ) 21 +ϵ t C 34 +ϵ · 1 1 s 1 2 1
C1
C1
j 3
1
≪ δ t +2 (log N )s + c (α, ϵ)(log N ) 2 s+ 2 ϵ s (log N )
1 +ϵ t 2
1
·
1 −ϵ 4
.
C1 Hence (note that by condition C7 we have C1 > (log N )20(t +s) )
1 (log N )t +1 t +2 3 +ϵ t s s+ 12 ϵ s 2 2 δ ( log N ) + c (α, ϵ)( log N ) ( log N ) · ≤ 9 δ t +1 ≪ c (α, ϵ)(log N )
t +s−1
1 1 −ϵ 4
C1
.
To finish the proof of the theorem it remains to estimate
:=
e2π iL1,j (n,α)
n∈Zt \{0} C7 ,C9
1=(δ1 ,...,δt +1 )∈{0,1}t +1
j
10
(−1)δ1 +···+δt +1
Λ(θ, j )
n1 · · · nt (nb(j )α − nt +1 )
g (n, j ).
For estimating 10 we need a generalization of Beck’s ‘Key Lemma’ provided in [2]. We define in analogy to [2, Section 7]: Definition 1. For given 1, j and l = (l1 , . . . , lt +1 ) ∈ Nt0+1 satisfying
(log N )20(t +s) ≤ 1 +
lτ
1
(log N )
2
<
N (j )2
for τ = 1, . . . , t
4
(17)
and 20(t +s)
(log N )
≤ 1+
1
lτ +1
(log N )2
<
N (j )2 4
t
,
(18)
we say that l is an ϵ-big-vector for b(j )α if
± U (l , j , ϵ) 2π iL1,j (n,α) 2π iL1,j (n,α) e − e , ≥ log N n∈U (l ,j ,ϵ+ ) n∈U (l ,j ,ϵ− ) where U ± (l , j , ϵ) := U (l , j , ϵ+ ) ∪ U (l , j , ϵ− ). Further we define the concept of a ‘special ϵ -line’ in the same way as it is done in [2], Definitions 7.2 and 7.3. For the sake of completeness we repeat the definitions here:
G. Larcher / Journal of Complexity 29 (2013) 397–423
421
Definition 2. Two integral vectors l = (l1 , . . . , lt +1 ) and h := (h1 , . . . , ht +1 ) satisfying the conditions as stated in (17) and (18) for l, are called ϵ-neighbors if
(1 + δ)ϵτ (hτ −lτ ) = (log N )9 for 1 ≤ τ ≤ t and, (1 + δ)(ht +1 −lt +1 ) = (log N )9(t +1) . ϵ
The notation l → h means that the ordered pair ⟨l , h⟩ of vectors satisfies the two conditions above. Definition 3. A sequence H = h(1) , h(2) , . . . of vectors satisfying the conditions as stated in (17) and (18) for l, is called a ‘special ϵ-line’ if
ϵ ϵ ϵ h(1) → h(2) → h(3) → . . . ,
that is, any two consecutive vectors in H are ϵ-neighbors. We will show a generalized form of Beck’s ‘Key Lemma’, namely. Lemma 8. For every α satisfying Lemma 6, (i.e., for almost all α) we have: For given 1 there is an N (α) such that for all N > N (α) and all j we have that every special ϵ-line contains at most one ϵ-big vector for b(j )α. Remark 3. It seems that from the Key Lemma of Beck as it is stated in [2], our generalized version of the Key Lemma, i.e., Lemma 8, follows immediately for almost all α. However, the formulation of Beck’s Key Lemma as it is stated in [2] is not quite complete. The complete formulation would be the following: ‘‘For any α ∈ Rt satisfying Lemma 4.2 (in [2]) there is an N (α), such that for all N ≥ N (α) every special ϵ-line contains at most one ϵ-big vector.’’ From this correct version our Lemma 8 does not follow immediately. Proof of Lemma 8. In the proof we can follow the proof of Beck’s Key Lemma almost line by line with the only two exceptions that we have to use our Lemma 6 (in suitable form) wherever (essentially two times) Beck uses his Lemma 4.2. We start with the proof of Lemma 8 until we have to use Lemma 6 for the first time. Further we show where to use Lemma 6 for the second time. The rest of the proof can be carried out quite analogously as it is done in the proof of the Key Lemma by Beck in [2]. So, for given 1 and j let H = h(1) , h(2) , . . . be a special ϵ-line with two ϵ-big vectors h(p) and h(q) for b(j )α. If
L1,j (n, α) ≤
1
for every n ∈ U ± (h(p) , j , ϵ)
(log N )2
(19)
then
1 − e2π iL1,j (n,α) ≤
1
(log N )2
for all n ∈ U ± (h(p) , j , ϵ)
(20)
and hence
2π iL1,j (n,α) 2π iL1,j (n,α) e − e n∈U (h(p) ,j ,ϵ+ ) n∈U (h(p) ,j ,ϵ− ) ± (p) U (h , j , ϵ) (p) + (p) − ≪ U (h , j , ϵ ) − U (h , j , ϵ ) + , (log N )2 and this by Lemma 6 with J = {j } is at most
± (p) 3+ 1 U (h , j , ϵ) 1 1 (p) 1 1 + 10 t ht +1 4 10 + 2 ¯ ¯ . ≪ c (α) (1 + δ) (log N ) (j1 · · · js ) 2 10 + (log N )2
(21)
422
G. Larcher / Journal of Complexity 29 (2013) 397–423
Further, by Lemma 6 and again with δ = (log1N )2 , C1 = (1 + δ)lt +1 +1 and C2 = (1 + δ)lt +1 we have
± (p) U (h , j , ϵ) ≫ (C1 − C2 )
1
(log N )2t
34 + 101 1 (p) 1 1 +1 t − c (α) (1 + δ)ht +1 (log N ) 2 10 (¯j1 · · · ¯js ) 2 + 10 34 + 101 1 (p) (p) 1 1 +1 t ≫ δ(1 + δ)ht +1 δ t − c (α) (1 + δ)ht +1 (log N ) 2 10 (¯j1 · · · ¯js ) 2 + 10 (p)
≫ δ t +1 (1 + δ)ht +1 for N ≥ N (α) (p) ht +1
≥ (log N )20(s+t ) . (p) h Therefore (again also since (1 + δ) t +1 ≥ (log N )20(s+t ) ) we have ± (p) U (h , j , ϵ) for N ≥ N (α). (21) ≪ (log N )2
since (1 + δ)
But this contradicts the assumption that h(p) is an ϵ-big vector for b(j )α. From here on we now proceed quite analogously as in the proof of the Key Lemma in [2], starting with formula (7.8) in Beck’s proof: The falsity of the assumption in (19) means, that for every α satisfying Lemma 6 there is an N (α) such that for all j we have that there is an n∗ ∈ U ± (h(p) , j , ϵ) such that
L1,j (n∗ , α) >
1
(log N )2
for all N ≥ N (α). In formula (7.23) Beck uses his Lemma 4.2 to estimate the maximal number of border points. At this place we again use our Lemma 6 with an implied constant c (α) not depending on j, quite analogously as we did at the beginning of this proof. This gives the proof of Lemma 8. Now we are ready to estimate We set
˜ (j , 1) := 10
n∈Zt \{0} C7 ,C9
10
and to finish the proof of the theorem:
e2π iL1,j (n,α) n1 · · · nt (nb(j )α − nt +1 )
g (n, j ).
This is just the sum 10 in the paper of Beck in [2] now with b(j )α instead of α. Beck’s 10 is estimated in the first part of Section 8 in [2] to be ≤ c (α)(log N )t log log N for all α satisfying Beck’s Lemma 4.2, by using Beck’s Key Lemma and Beck’s Lemma 4.2 (note that in [2] the symbol k stands for our symbol t). Quite analogously, by using our generalized Key Lemma (Lemma 8) instead of Beck’s Key Lemma, and using our Lemma 6 instead of Beck’s Lemma 4.2 we get
˜ 10 (j , 1) ≤ c (α)(log N )t log log N for all α satisfying Lemma 6 and for all j and with C (α) not depending on j.
(By using Beck’s auxiliary results we just would obtain ˜
10
(j , 1) ≤ c (α, j )(log N )t log log N,
i.e., an estimate with a constant c (α, j ) depending also on j.) Now of course, for all α satisfying Lemma 6, i.e., for almost all α, we have
≤ c (α)(log N )s+t log log N . 10 This finishes the proof of the theorem.
G. Larcher / Journal of Complexity 29 (2013) 397–423
423
Acknowledgment The author was supported by the Austrian Science Fund (FWF), Project P21196. References [1] J. Beck, A two-dimensional van Aardenne Ehrenfest theorem in irregularities of distribution, Compos. Math. 72 (1989) 269–339. [2] J. Beck, Probabilistic diophantine approximation, I. Kronecker sequences, Ann. of Math. 140 (1994) 451–502. [3] D. Bilyk, M.T. Lacey, On the small ball inequality in three dimensions, Duke Math. J. 143 (2008) 81–115. [4] D. Bilyk, M.T. Lacey, The supremum norm of the discrepancy function: recent results and connections, 2012, Preprint. [5] D. Bilyk, M.T. Lacey, A. Vagharshakyan, On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008) 2470–2502. [6] J. Dick, F. Pillichshammer, Digital Sequences, Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010. [7] D. Gomez-Perez, R. Hofer, H. Niederreiter, A general discrepancy bound for hybrid sequences involving Halton sequences, Unif. Distrib. Theory 8 (2013) 31–45. [8] R. Hofer, G. Larcher, On existence and discrepancy of certain digital Niederreiter–Halton sequences, Acta Arith. 141 (2010) 369–394. [9] R. Hofer, G. Larcher, Metrical results on the discrepancy of Halton–Kronecker sequences, Math. Z. 271 (2012) 1–11. [10] A. Khintchine, Ein Satz über Kettenbrüche mit arithmetischen Anwendungen, Math. Z. 18 (1923) 289–306. [11] A. Khintchine, Kettenbrüche, B.G. Teubner Verlagsgesellschaft, Leipzig, 1956. [12] G. Larcher, H. Niederreiter, Generalized (t , s)-sequences, Kronecker-type sequences, and diophantine approximation of formal Laurent series, Trans. Amer. Math. Soc. 347 (1995) 2051–2073. [13] H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987) 273–337. [14] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, in: CBMS-NSF Series in Applied Mathematics, vol. 63, SIAM, Philadelphia, 1992. [15] H. Niederreiter, On the discrepancy of some hybrid sequences, Acta Arith. 138 (2009) 373–398. [16] H. Niederreiter, Further discrepancy bounds and an Erdös–Turan–Koksma inequality for hybrid sequences, Monatsh. Math. 161 (2010) 193–222. [17] H. Niederreiter, Quasi-Monte Carlo methods, in: R. Cont (Ed.), Encyclopedia of Quantitative Finance, Wiley, Chichester, 2010, pp. 1460–1472. [18] H. Niederreiter, Improved discrepancy bounds for hybrid sequences involving Halton sequences, Acta Arith. 155 (2012) 71–84. [19] W.M. Schmidt, Metrical theorems of fractional parts of sequences, Trans. Amer. Math. Soc. 110 (1964) 493–512. [20] J. Spanier, Quasi-Monte Carlo methods for particle transport problems, in: H. Niederreiter, P.J.-S. Shiue (Eds.), Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, in: Lecture Notes in Statistics, vol. 106, Springer, New York, 1995, pp. 121–148.
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Journal of Complexity journal homepage: www.elsevier.com/locate/jco
Probabilistic diophantine approximation and the distribution of Halton–Kronecker sequences Gerhard Larcher Institute of Financial Mathematics, University of Linz, Altenberger Strasse 69, 4040 Linz, Austria
article
info
Article history: Received 1 February 2013 Accepted 7 May 2013 Available online 21 May 2013 Keywords: Uniform distribution of sequences Discrepancy Diophantine approximation Kronecker sequences Halton sequences
abstract By a Halton–Kronecker sequence we mean a sequence in the s + t-dimensional unit-cube which is the combination of an s-dimensional Halton sequence and a t-dimensional Kronecker sequence ({n · α})n=0,1,... with α ∈ Rt . The investigation of such ‘hybrid sequences’ for their use in Monte Carlo and quasi-Monte Carlo methods first was motivated by Spanier (1995) [20]. By suitably adapting techniques of Jozsef Beck on probabilistic diophantine approximation, developed in Beck (1994) [2], we can show that for almost all α ∈ Rt for the discrepancy DN of a Halton–Kronecker (log N )s+t +ϵ
) for all ϵ > 0, which most sequence we have DN = O( N probably essentially is the best possible metrical result for this type of sequences. © 2013 Elsevier Inc. All rights reserved.
1. Introduction Let (zn )n≥0 be a sequence in the d-dimensional unit-cube [0, 1)d . Then the discrepancy of the first N points of the sequence is defined by
AN (B) DN = sup − λ(B) , N d B⊆[0,1)
where AN (B) := #{n : 0 ≤ n < N , zn ∈ B},
λ is the d-dimensional volume and the supremum is taken over all axis-parallel subintervals B ⊆ [0, 1)d . The sequence (zn )n≥0 is called uniformly distributed if limN →∞ DN = 0.
E-mail address: [email protected]. 0885-064X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jco.2013.05.002
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G. Larcher / Journal of Complexity 29 (2013) 397–423
It is the most well-known conjecture in the theory of irregularities of distribution, that for every sequence (zn )n≥0 in [0, 1)d we have DN ≥ cd ·
(log N )d
N for a constant cd > 0 and for infinitely many N. Hence sequences whose discrepancy satisfies DN
= O
(log N )d
N
are called ‘low-discrepancy
sequences’. Note that recent investigations of Bilyk, Lacey et al., see for example [3] or [5], have led some people to conjecture that
(log N )
d+1 2
N
instead of
(log N )d N
is the best possible order for the discrepancy of sequences
in [0, 1) . At the moment the best known general lower bound for the discrepancy of sequences in [0, 1)d is d
d
DN ≥ cd ·
(log N ) 2 +ϵ(d)
N for infinitely many N, with some small ϵ(d) > 0. For more details on this topic see [1] or [4]. There are three groups of ((almost) low-discrepancy) sequences which are of main interest for applications in quasi-Monte Carlo methods. (Here by a quasi-Monte Carlo method we mean simulation in the setting of Monte Carlo methods, but using deterministic, i.e., quasi-random (usual low-discrepancy) point sets instead of pseudo-random point sets.) Indeed, these are (until now) the only known types of sequences containing concrete examples of (almost) low-discrepancy sequences. These are Kronecker sequences, Halton sequences and (T, s)-sequences in the sense of Niederreiter. The most classical type are the Kronecker sequences. A Kronecker sequence is of the form zn = ({n · α})n=0,1,... = (({n · α1 } , . . . , {n · αd }))n=0,1,... for some α = (α1 , . . . , αd ) ∈ [0, 1)d . The sequence is uniformly distributed in [0, 1)d iff 1, α1 , . . . , αd are linearly independent over Z. It was shown by Beck in his ingenious paper [2], that for almost all α in [0, 1)d and all ϵ > 0 for the discrepancy of the Kronecker sequence we have
DN = O
(log N )d+ϵ
N
.
He moreover showed that this metrical result essentially is best possible. (Indeed Beck gave an even more precise result than stated above.) So almost every Kronecker sequence is an ‘almost’ lowdiscrepancy sequence. (For d = 1 the result of Beck already was shown by Khintchine [10]. For d ≥ 2 Beck improved a result of W.M. Schmidt [19], who had given the bound DN = O
(log N )d+1+ϵ N
for almost all α .)
Halton sequences are defined as follows: Let b1 , . . . , bd ≥ 2 be pairwise coprime integers, then the Halton sequence (zn )n=0,1,... with (1)
(d)
zn = zn , . . . , zn
, in bases b1 , . . . , bd is given by
zn(j) := ψbj (n) where
ψbj (n) :=
∞
i−1 ni b − j
i =0
for n with base bj representation n=
∞
ni bij .
i=0
It is easy to show that every Halton sequence is a low-discrepancy sequence.
G. Larcher / Journal of Complexity 29 (2013) 397–423
399
The concept of (t , s)-sequences in a base b and of (T, s)-sequences in a base b (see for example [13,12,14], or [6]) was introduced by Niederreiter and it contains also earlier special examples of sequences of the same type considered by Sobol and by Faure. We do not need to explain this concept here, since we will not deal with it in this paper. We just note that it provides many concrete examples of low-discrepancy sequences. In the last years there grew considerable interest in the distribution of ‘hybrids’ of such sequences, i.e.: take an s-dimensional sequence (xn )n=0,1,... of a certain type and a t-dimensional sequence (yn )n=0,1,... of an other type and combine them to an s + t-dimensional hybrid sequence
(zn )n=0,1,... := ((xn , yn ))n=0,1,... . The use of such hybrid sequences in Monte Carlo and in quasi-Monte Carlo methods first was suggested by Spanier [20]. For the new concept of hybrid-Monte Carlo based on hybrid sequences, see for example also [17] or [7]. It is rather obvious that in general the hybrid of two low-discrepancy sequences is no longer a low-discrepancy sequence (see for example [8] where – for instance – it was shown that the hybrid of a Halton sequence in bases b1 , . . . , bd and a certain most basic low-discrepancy (t , s)-sequence in a different base b no longer is a low-discrepancy sequence). Hybrids of the two classical types of sequences, namely of Halton sequences and of Kronecker sequences – we will call them Halton–Kronecker sequences – were first studied by Niederreiter in [15,16] and in [18]. They are uniformly distributed if the Kronecker-component of the hybrid is uniformly distributed. From a metrical point of view the discrepancy of Halton–Kronecker sequences was studied in [9]. It was shown there: For every Halton sequence in [0, 1)s and almost every α ∈ Rt for the discrepancy of the s + tdimensional Halton–Kronecker sequence we have
DN = O
(log N )s+t +1+ϵ
N
(1)
for every ϵ > 0. In the case t = 1 we even have
DN = O
(log N )s+1+ϵ
(2)
N
for every ϵ > 0. To obtain these results we adapted the techniques of Khintchine [10] and W.M. Schmidt [19]. It is the aim of this paper to improve (1) to its probably (essentially) best possible form, i.e., to show Theorem 1. For every Halton sequence in [0, 1)s and almost every α ∈ Rt for the discrepancy of the s + t-dimensional Halton–Kronecker sequence we have
DN = O
(log N )s+t +ϵ
N
(3)
for every ϵ > 0. The proof of the theorem heavily depends on the deep techniques on probabilistic diophantine approximation provided by Beck in his ingenious paper [2]. Remark 1. In [2] Beck also gave a lower bound from which for example it follows that for almost all
α ∈ [0, 1)d for the discrepancy DN of the d-dimensional Kronecker sequence we have (log N )d DN = Ω . N
400
G. Larcher / Journal of Complexity 29 (2013) 397–423
At the moment it seems to be out of reach to give an analogous metrical lower bound for Halton– Kronecker sequences, at least in the cases where s ≥ 2, since untilnow for s ≥ 2 it is not even known
if for the s-dimensional Halton sequence alone we have DN = Ω
(log N )s N
.
Remark 2. Until now we do not know any concrete example of a (almost) low-discrepancy Halton– Kronecker sequence. Even in the most simple case, where s = t = 1, we do not have any such concrete example. So for instance it would be of great interest to study the discrepancy DN of
√ (zn )n=0,1,... = (xn , {n · 2})
n=0,1,...
,
where xn is the one-dimensional Halton sequence in base 2, i.e., the van der Corput sequence, and to check whether
DN = O
(log N )2+ϵ
N
holds or not. The paper is organized as follows: In Section 2 we prepare the prerequisites to be able to handle the discrepancy of Halton–Kronecker sequences with the help of the Poisson-summation formula. In Section 3 we give necessary auxiliary results from probabilistic diophantine approximation, most of which are suitable generalizations of the results of Beck given in Sections 4 and 9 in [2]. In Section 4 we finish the proof of our Theorem 1. During the proof we will very often refer to the paper of Beck [2]. It is quite clear that writing our paper would not have been possible without the ingenious techniques developed by Beck in [2]. 2. Prerequisites We consider the d = s + t-dimensional sequence
(zn )n=0,1,... = ((xn , yn ))n=0,1,... , where (xn )n=0,1,... is the s-dimensional Halton sequence in pairwise coprime bases b1 , . . . , bs and (yn )n=0,1,... is the t-dimensional Kronecker sequence yn = ({n · α})n=0,1,... = (({n · α1 } , . . . , {n · αt }))n=0,1,... . Let I = [0, β) × [0, γ) ⊆ [0, 1)d with β = (β1 , . . . , βs ) and γ = (γ1 , . . . , γt ). We will choose in the following certain disjoint subsets Iint and Ibor of [0, 1)d such that Iint ⊆ I ⊆ Iint ∪ Ibor . Then with AN (I ) := #{n : 0 ≤ n < N , zn ∈ I } we obviously have
|AN (I ) − N λ(I )| ≤ |AN (Iint ) − N λ(Iint )| + AN (Ibor ) + N λ(Ibor ). We choose Iint as follows: Let
βi = (i)
with βj
β1(i) b1i
+
β2(i) b2i
+ ···
∈ {0, 1, . . . , bi − 1}.
(4)
G. Larcher / Journal of Complexity 29 (2013) 397–423
401
Then let I (j1 , . . . , js , k1 , . . . , ks , γ) :=
j −1 s i β (i) l
i=1
l=1
j i −1 βl(i) ki
+
bli
j
bii
,
l =1
+
bli
ki + 1
j
bi i
× [0, γ)
for positive integers j1 , . . . , js and ki ∈ {0, 1, . . . , bi − 1} for i = 1, . . . , s. By the construction of the Halton sequence there is a unique
j
j
r = r (j1 , . . . , js , k1 , . . . , ks ) ∈ 0, 1, . . . , b11 b22 · · · bjss − 1 such that zn ∈ I (j1 , . . . , js , k1 , . . . , ks , γ) if and only if j
n ≡ r mod (b11 · · · bjss ) and yn ∈ [0, γ) .
(5)
For x ∈ R let [x] denote the largest integer less or equal x. Then let Li := logbi N + 1 and define Iint as union of disjoint intervals by
Iint :=
L1
β
···
j 1 =1
(1)
(s)
β j −1
−1
j1 Ls
···
js =1 k1 =0
s
I (j1 , . . . , js , k1 , . . . , ks , γ).
ks =0
Further let Ibor :=
s
i =1
i−1
L Li s i βl(i) 1 βl(i) , + L × [0, 1) × [0, γ). [0, 1) × l l
j =1
l =1
bi
l =1
bi
bi i
j=i+1
Then indeed we have Iint ⊆ I ⊆ Iint ∪ Ibor , and by (4) and (5):
|AN (I ) − N λ(I )| 1 #{m | 0 ≤ n = r (j , k ) + mb(j ) < N and {nα} ∈ [0, γ)} − N ≤ λ([0, γ)) j ,k b( j ) s s 1 1 + N L +1 +N L i =1
bi i
i =1
bi i
≤ #{0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j , k ), γ + θ(j , k ))} j k s − N (j )λ ([θ(j , k ), γ + θ(j , k ))) + 1+ 3, i =1 j k where – – – – – –
# means the number of elements of a set, j = (j1 , . . . , js ), k = (k1 , . . . , ks ), θ(j , k ) := −r (j , k ) · α, intervals always are taken modulo one in each coordinate, L1 Ls j 1 =0 · · · j always means summation j s =0 ,
402
–
G. Larcher / Journal of Complexity 29 (2013) 397–423
k
means summation j b11
– b(j ) := – N (j ) :=
j b22 N b(j )
· · ·
j bss ,
βj(11) −1 k1 =1
···
βj(ss) −1 k s =1
,
and
.
Since
j
1≤
s
bi (logbi N + 1),
i =1
k
we have shown that to prove Theorem 1 it suffices to show that for almost all α for all ϵ > 0 we have
|# {0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j ), γ + θ(j ))} − N (j )λ [θ(j ), γ + θ(j ))|
j
≤ c (α, ϵ, s, t , b1 , . . . , bs ) ·
(log N )s+t +ϵ
(6)
N
for all N, for all reals θ(j ) and all γ . Now it would be tempting to conclude that the assertion of Theorem 1 of our paper is a direct consequence of the above fact and of Theorem 1 in [2], since by Beck’s theorem it follows that for all j, for almost all α we have for all γ , all θ and for all ϵ > 0 the estimate
|# {0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j ), γ + θ(j ))} − N (j )λ[θ(j ), γ + θ(j ))| (log N )t +ϵ =O N
(7)
holds for all N, and that hence for almost all α we have
|# {0 ≤ m < N (j ) | {mb(j )α} ∈ [θ(j ), γ + θ(j ))} − N (j )λ [θ(j ), γ + θ(j ))|
j
=O
(log N )t +s+ϵ
N
,
(8)
with a big-O constant depending on α and on ϵ . However, of course, there is a substantial technical ‘uniformity problem’ here: Beck’s theorem guarantees that for almost every α ∈ [0, 1)d there is a constant c (α, j ) in the bigO-estimate of (7). But this constant is not necessarily bounded in j, so that argument (8) cannot be used. So we have to go through the proof of Beck and to modify and adapt his proof for our more general situation. For given j and θ(j ) = (θ1 (j ), . . . , θt (j )) we consider in analogy to [2], formula 3.11 the ‘roof-likeaverage’
¯ j (α, x; N ) := D
N (j )2
t
2
·
1
2 −
2 N (j )2 2 N (j )2
2 N (j )2
··· −
2 N (j )2
2
t
1−
−2 τ =1
+ θ(j ), x + u + θ(j ); ut +1 , N (j ) + ut +1 ) 1 − with u := (u1 , . . . , ut ), x := (x1 , . . . , xt ), and where for v = (v1 , . . . , vt ), y = (y1 , . . . , yt ) ∈ [0, 1)t
N (j )2 2
|ut +1 | 2
|uτ | Dj (b(j )α; u
du1 · · · dut dut +1 ,
(9)
G. Larcher / Journal of Complexity 29 (2013) 397–423
403
we define
D(b(j )α; v , y ; a, b) :=
1 − (b − a)
n
(yτ − vτ ),
τ =1
n
where
t
here means summation over all integers n with a < n ≤ b for which
v ≤ nb(j )α ≤ y mod [0, 1)t . Analogously to [2, 3.12.], by using Poisson’s summation formula we obtain that formally
t 1 − e2π inτ xτ
¯ j (α, x; N ) = it −1 D
n∈Zt +1 \{0}
×
1−e
τ =1
sin 2π Nn(jτ)2
2
2π Nn(jτ)2
2π i(b(j )(n1 α1 +···nt αt )−nt +1 )N (j)
2π (b(j )(n1 α1 + · · · nt αt ) − nt +1 )
×
2π nτ
sin 2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 )
2 t
2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 )
e2π iθτ (j ) .
(10)
τ =1
Here and in the following we always have n = (n1 , . . . , nt ) if n ∈ [0, 1)t , resp. n = (n1 , . . . , nt +1 ) if n ∈ [0, 1)t +1 . We set 1−e
2π inτ xτ
2π nτ
= −ixτ
and
sin 2π Nn(jτ)2
2π Nn(jτ)2
2 = 1 if nτ = 0.
The last additional factor τ =1 e2π iθτ (j ) in (10) compared with [2, 3.12] comes from the fact that we have to consider intervals [u + θ(j ), x + u + θ(j )) instead of [u, x + u) in [2]. In formulas (5.3)–(5.10) in [2] it is shown that for given j for almost , 1)t the series in t all 2απ iθ∈τ (j[0 ) (10) is absolutely convergent for all x and all N. (The additional factor τ =1 e of course does not influence this fact.) Hence for almost all α ∈ [0, 1)t the series is absolutely convergent for all j , N (j ), θ(j ) and all x, and therefore for almost all α indeed we have
t
¯ j (α, x; N ) = it −1 D
j
j
×
t 1 − e2π inτ xτ
n∈Zt +1 \{0} τ =1
2π nτ
sin 2π Nn(jτ)2 2π Nn(jτ)2
2
1 − e2π i(b(j )(n1 α1 +···nt αt )−nt +1 )N (j) 2π (b(j )(n1 α1 + · · · nt αt ) − nt +1 )
×
sin 2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 ) 2π (b(j )(n1 α1 + · · · + nt αt ) − nt +1 )
2 t
e2π iθτ (j ) .
(11)
τ =1
We will show in the following sections that for almost all α ∈ [0, 1)t for every ϵ > 0 there is a constant c (α, ϵ, s, t , b1 , . . . , bs ) such that
¯ j (α, x; N ) ≤ c (α, ϵ, s, t , b1 , . . . , bs ) D
j
holds for all N , θ(j ) and all x.
(log N )s+t +ϵ N
(12)
404
G. Larcher / Journal of Complexity 29 (2013) 397–423
From (12) we derive (6) and therefore the assertion of Theorem 1 then in the following way: The left hand side of (6) is
D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )).
j
We consider α such that each coordinate of α satisfies property (13) of Lemma 1 (see Section 3). Almost all α are of this form. For these α we have for all j , θ(j ), all u = (u1 , . . . , ut ) with |uτ | ≤
2 N (j )2
for all τ = 1, . . . , t
and all ut +1 with |ut +1 | ≤ 2:
|D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )) − D(b(j )α; u + θ(j ), u + θ(j ) + γ; ut +1 , ut +1 + N (j ))| ≤ 4 + |D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )) − D(b(j )α; u + θ(j ), u + θ(j ) + γ; 0, N (j ))| t 1 ≤4+ # 0 ≤ n ≤ N (j ) {nb(j )ατ } ∈ θ (j )τ , θ (j )τ + N (j )2 τ =1 1 { } + # 0 ≤ n ≤ N (j ) nb(j )ατ ∈ θ (j )τ + γτ , θ (j )τ + γτ + N (j )2 t 1 ≤ 4 + 2t + 2 # 0 ≤ n < N (j ) ∥nb(j )ατ ∥ ≤ , N (j )2 τ =1
and hence
¯ Dj (α, x; N ) − D(b(j )α; θ(j ), θ(j ) + γ; 0, N (j )) j j t # 0 ≤ n < N (j ) ∥nb(j )ατ ∥ ≤ ≤ (4 + 2t ) + 2 j
j
τ =1
1
N (j )2
≤ c (α, ϵ, t )(log N )s+1+ϵ for every ϵ > 0 by Lemma 1. Hence our Theorem 1 follows from (12). So Sections 3 and 4 will be devoted to the proof of (12). 3. Auxiliary lemmas In this section we prove some auxiliary lemmas. Here Lemmas 3–6 essentially are adapted versions of some of the lemmas in Sections 4 and 9 in [2]. Lemma 1, which already was used in Section 2, is an easy consequence of Lemma 2. This Lemma 2 already was stated and proved in [9, Proposition 2]. Since the proof given there contained a certain gap, we use the opportunity to give a complete proof here. (Here and in the following by ‘≪’ we always mean ‘
1 ≤ C (α, ϵ)(log N )s+1+ϵ . # 0 ≤ n < N (j ) ∥nb(j )α∥ < N (j )2
j
We remind that, as already defined in Section 2, by by b(j ) we mean b(j ) =
j b11
···
j bss ,
j
we mean
and by N (j ) we mean [
N b(j )
].
L1
j 1 =1
···
(13)
Ls
js =1
with Li = [logbi N ] + 1,
G. Larcher / Journal of Complexity 29 (2013) 397–423
405
As mentioned above, for the proof of Lemma 1 we need. Lemma 2. For x ∈ R let ak (x) denote the k-th continued fraction coefficient of x. For any L ∈ N let SL (α) :=
L
···
j 1 =1
L L
ak (b(j )α).
js =1 k=1
Then for almost all α ∈ R we have for every ϵ > 0 that SL (α) = O(Ls+1+ϵ ), where the implied constant is independent of L. Proof of Lemma 1. For given j let k(j ) be such that qk(j )−1 (b(j )α) < N (j ) ≤ qk(j ) (b(j )α), where qk (x) is the k-th best approximation denominator of x. Then, since
1 b(j )α − pk(j )−1 ≥ qk(j )−1 2qk(j ) · qk(j )−1 we have
# 0 ≤ n < N (j ) ∥nb(j )α∥ <
1 N (j )
≤
2
1
·
N (j )
2
1 1 2qk(j ) ·qk(j )−1
≪ ak(j ) .
So by Lemma 2
j
# 0 ≤ n < N (j ) ∥nb(j )α∥ <
for almost all α.
1 N (j )
2
≪ Slog N (α) = O (log N )s+1+ϵ
Proof of Lemma 2. For C ∈ N let EC ⊆ [0, 1) be EC := α ∈ [0, 1) | ak (b(j )α) < C (¯j1 · · · ¯js k)4 for all j1 , . . . , js ∈ N0 and all k ∈ N ,
where ¯j := max(1, j). We first show that lim λ(EC ) = 1.
C →∞
The complement E¯ C of EC contains all α for which there exist k0 and j (0) such that (0)
ak0 (b(j (0) , α)) ≥ C (¯j1 · · · ¯j(s0) k0 )4 . (0)
(0)
For such α denote by k0 (α) the minimal such k0 , by j1 (α) the minimal such j1 for the given k0 (α), (0)
(0)
(0)
further j2 (α) the minimal j2 for the given k0 (α) and j1 (α), and so on. Then E¯ C =
∞ ∞ k0 =1 (0) j 1 =0
···
∞
α ∈ E¯ C | k0 (α) = k0 , j(10) (α) = j(10) , . . . , j(s0) (α) = j(s0) .
(0) j s =0
We consider the measure of the set (0) (0) S˜ := α ∈ E¯ C | k0 (α) = k0 , j1 (α) = j1 , . . . , j(s0) (α) = j(s0) .
406
G. Larcher / Journal of Complexity 29 (2013) 397–423
˜ then for the k0 − 1-st convergent If α ∈ S,
pk −1 0 qk −1 0
of b(j (0) )α we have
(0) 1 1 b(j )α − pk0 −1 < < . 2 (0) (0) 2 ¯ qk0 −1 qk0 −1 ak0 qk0 −1 (j1 · · · ¯js k0 )4 C Let α ∈ S˜ be such that this qk0 −1 = q. Then
α −
< ( 0 ) qb(j )
1
a
q2 b
(j (0) )(¯j(0) 1
· · · ¯j(s0) k0 )4 C
,
for some a ∈ {0, . . . , qb(j (0) )}, i.e., α is contained in a set of volume Since
2 (0) (0) q(¯j1 ···¯js k0 )4 C
.
(0)
q = qk0 −1 < (a1 + 1) · · · (ak0 −1 + 1) ≤ 2k0 (a1 · · · ak0 −1 ) ≤ (2C ¯j1 · · · ¯j(s0) k0 )k0 , we have (0) (0) (2C ¯j1 ···¯js k0 )k0
λ(S˜ ) < ≪
2
q=1
q(¯j1 · · · ¯js k0 )4 C
(0)
≪
(0)
log C
1
C
(¯j(10) · · · ¯j(s0) k0 )2
1
(¯j(10) · · · ¯j(s0) k0 )4 C
(0)
k0 log(C ¯j1 · · · ¯j(s0) k0 )
.
Hence λ(E¯ C ) ≪ C and the assertion limC →∞ λ(EC ) = 1 follows. Next we want to show that we have log C
ak (b(j )α)dα ≤ B(s, C ) log(kj1 · · · js ), EC
where B(s, C ) is a constant depending only on s and on C . We define ‘elementary intervals of order h’ in [0, 1) by ,...,js ) Ig(1j1,..., gh := {α ∈ [0, 1) | ai (b(j )α) = gi for all i ∈ {1, . . . , h}} .
From the fact that ψn : [0, 1) → [0, 1), x → {nx} is measure-preserving (in the sense of Lebesguemeasure) for every n ∈ N we know for all j1 , . . . , js ∈ N0 and for all h ∈ N, g1 , . . . , gh ∈ N that ,...,js ) (0,...,0) λ(Ig(1j1,..., gh ) = λ(Ig1 ,...,gh ),
and together with [11, Eq. (57)] we derive for all j1 , . . . , js ∈ N0 that ,...,js ) λ(Ig(1j1,..., gh−1 ,gh ) =
2 gt2
,...,js ) λ(Ig(1j1,..., gh−1 ).
Hence, and by the definition of EC we have
ak (b(j )α)dα ≤ EC
=
C (¯j1 ···¯js k)4
∞
gh =1
g1 =1
C (¯j1 ···¯js k)4
∞
gh =1
g1 =1
C (¯j1 ···¯js k)4
=
gh =1 C (j1 ···js k)4
=
gh =1
···
gh−1 =1
···
∞ 2
gh 2 gh
∞
g1 =1
∞
(j ,...,js ) Ig 1,...,g 1
ak (b(j )α)dα
h
,...,js ) gh λ(Ig(1j1,..., gh−1 ,gh )
gh−1 =1
···
∞
,...,js ) λ(Ig(1j1,..., gh−1 ,gh )
gh−1 =1
≤ B(s, C ) log(¯j1 · · · ¯js k).
G. Larcher / Journal of Complexity 29 (2013) 397–423
407
So we have
L
L L
···
EC j =1 1
j s =1 k =1
ak (b(j )α)
ϵ 1+ s+ 1
(¯j1 · · · ¯js k)
dα ≤
L
···
j1 =1
L L B(C , s) log(k¯j1 · · · ¯js ) ϵ
js =1 k=1
(¯j1 · · · ¯js k)1+ s+1
dα
≪ const (C , ϵ). Since the integrand above is monotonically increasing in L, we have that for almost all α ∈ EC it is L
···
j1 =1
L L
ak (b(j )α)
js =1 k=1
(¯j1 · · · ¯js k)1+ s+1
ϵ
dα ≪ const (C , ϵ)
and hence SL (α) ≪ const (C , ϵ)Ls+1+ϵ for all L ∈ N.
The next lemma is the analogue to Lemma 4.4 in [2]. Lemma 3. For almost all α ∈ [0, 1)t for all ϵ > 0 there is a c (α, ϵ) such that
j
(∥nb(j )α∥)
t
−1
n¯ τ
¯ |n1 · · · nt |)t +2 (log
≤ c (α, ϵ)(log N )s+ϵ
τ =1
n∈Zt \{0}
for all N. ¯ := max(1, log |x|). Here and in all the following n¯ τ := max(1, |nτ |) and logx Proof of Lemma 3. By Lemma 3 in [9] we know that for almost all α ∈ [0, 1)t there is a c (α) > 0 such that
∥nb(j )α∥ > c (α)(|¯n1 · · · n¯ t | · (¯j1 · · · ¯js ))−3 for all j and all n ∈ Zt \ {0}, i.e., 1
|log ∥nb(j )α∥ |
≫
c (α) . ¯ log(|n1 · · · nt | · (¯j1 · · · ¯js ))
(14)
In analogy to Schmidt [19] we consider the integral
Jδ (j , n) :=
[0,1)t
−1 ∥nb(j )α∥ · |log ∥nb(j )α∥ |1+δ dα,
which has a finite value c (δ) for any δ > 0, that is independent of j and n. Hence with δ := t +1 1 and for every ϵ > 0 we have
[0,1)t
−1 ∥nb(j )α∥ · |log ∥nb(j )α∥ |1+δ j
n∈Zt \{0}
×
t
−1 1+ ϵ ϵ 1+ 2s 2s ) · ¯j1 · · · ¯js dα
¯ τ) (¯nτ (logn
1+δ
τ =1
=
j
n∈Zt \{0}
Jδ (j , n)
t
τ =1
−1 1+ ϵ ϵ 1+ 2s 1+δ 2s ¯ ¯ ¯ (¯nτ (log nτ ) ) · j1 · · · js ≪ c (ϵ),
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G. Larcher / Journal of Complexity 29 (2013) 397–423
for all N, and therefore for almost all α there is a c (α, ϵ) such that
−1 ∥nb(j )α∥ · |log ∥nb(j )α∥ |1+δ n∈Zt \{0}
j
×
t
−1 1+ ϵ ϵ 1+ 2s 2s · · · ¯js < c (α, ϵ) ) · ¯j1
¯ nτ ) (¯nτ (log
1+δ
τ =1
for all N. By (14) therefore for almost all α there is a c˜ (α, ϵ) such that
∥nb(j )α∥ ·
ϵ ϵ 1+ 2s 1+ 2s 1+δ ¯ ¯ ¯ (¯nτ (log nτ ) ) · j1 · · · js
τ =1
n∈Zt \{0}
j
t
−1 ¯ (|n1 · · · nt | · (¯j1 · · · ¯js ))) × (log
≤ c˜ (α, ϵ),
1+δ
for all N. Hence, using that
¯ |n1 · · · nt | · log ¯j1 · · · log ¯js , ¯ (|n1 · · · nt | · (¯j1 · · · ¯js ))1+δ ≪ log log for almost all α there is a c¯ (α, ϵ) such that
j
∥nb(j )α∥ ·
t
(t +1)(δ+1)
¯ |n1 · · · nt |) (log
n¯ τ
τ =1
n∈Zt \{0}
s
1+ ϵ ji s
−1
¯
≤ c¯ (α, ϵ)
i =1
for all N, and therefore, for almost all α and all ϵ > 0 we have
j
∥nb(j )α∥ ·
−1
n¯ τ
(t +2)
¯ |n1 · · · nt |) (log
≪ c¯ (α, ϵ)(log N )s+ϵ
τ =1
n∈Zt \{0}
for all N.
t
The following Lemma 4 is the generalization of Lemma 4.1 in [2], however in a slightly weaker form. Lemma 4. For almost all α ∈ Rt for every ϵ > 0 there is a c (α, ϵ) such that
j
∥nb(j )α∥ ·
n∈Zt \{0} C11 ,C12
t
−1 n¯ τ
≤ c (α, ϵ)(log N )s+t +ϵ
τ =1
for all N. Here the conditions C11 and C12 mean C11 : nτ ≤ N 3
for τ = 1, . . . , t .
C12 :
t
n¯ τ
· ∥nb(j )α∥ < (log N )20(s+t ) .
τ =1
Proof of Lemma 4. For the proof of Lemma 4 we will need Lemma 5.
G. Larcher / Journal of Complexity 29 (2013) 397–423
409
Let α ∈ Rt be one of the almost all α satisfying Lemma 5(a) and (b), let ϵ > 0 and c˜ (α, ϵ) like in Lemma 5(b) (note there is a second constant c¯ different from c˜ in the expression below), then:
j
∥nb(j )α∥ ·
t
−1 n¯ τ
τ =1
n∈Zt \{0} C11 ,C12
l¯2 · · · l¯2t 1
c¯ log log N
≤
l ∈ Nt 0 l 3 3 τ ≤N 3 τ =1,...,t
j
v=−˜c (α,ϵ) log log N n∈l-box
2v−1
3l1 +···lt . (15)
C13
Here we have to explain what we mean, by an l-box, by condition C13 in (15) by c¯ , and why we can choose the bounds for v in this way: For given l = (l1 , . . . , lt ) ∈ Nt0 by the l-box we mean the set
lτ −1
n ∈ Zt | 33
lτ ≤ n¯ τ < 33 for τ = 1, . . . , t ,
lτ −1
where we set 33 = 1 if lτ = 0, and ¯l := max(1, l). By condition C13 we mean C13 :
2v−1
¯ · · ·¯
3l1 +···+lt l21
l2t
≤ ∥nb(j )α∥ ·
t
n¯ τ
2v
<
¯ · · · ¯l2t
3l1 +···+lt l21
τ =1
.
Note that 2v
¯ · · ·¯
3l1 +···+lt l21
l2t
≪
2v
¯ n¯ 1 · · · log ¯ n¯ t · ¯l21 · · · ¯l2t log
therefore by Lemma 5(b) the inner sum
n∈l-box C13
,
would be an empty sum for v < −˜c (α, ϵ) log log N.
Finally c¯ is determined such that 2c¯ log log N 3l1 +···+lt ¯l21 · · · ¯l2t
> (log N )20(s+t ) .
This is possible since 2c¯ log log N 3l1 +···+lt ¯l21 · · · ¯l2t
≫
2c¯ log log N
(log N )t (log log N )2t
> (log N )20(s+t )
for c¯ large enough in dependence only on s and t. Hence the inner sum
empty sum for v > c¯ log log N. We proceed with the estimation of the right hand side of (15) which is c¯ log log N
≤
l ∈Nt 0 l 33 τ ≤N 3 τ =1,...,t
¯l21 · · · ¯l2t
v=−˜c (α,ϵ) log log N
2v−1
3l1 +···lt
j
1,
n∈l-box C13
and by Lemma 5(a) – with c (α, ϵ) like in this Lemma 5(a) – this is at most c¯ log log N
l ∈Nt 0 l 3 3 τ ≤N 3 τ =1,...,t
v=−˜c (α,ϵ) log log N
¯l21 · · · ¯l2t 2v−1
3l1 +···lt · c α,
ϵ
ϵ 2
ϵ
· 2v c˜ (α, ϵ) log log N max(1, v)1+ 2 (log N )s+ 2
n∈l-box C13
in (15) would be the
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G. Larcher / Journal of Complexity 29 (2013) 397–423
ϵ ϵ ϵ ¯l21 · · · ¯l2t 3l1 +···+lt (log N )s+ 2 (log log N )3+ 2 · ≪ c ′ α, 2
l ∈ Nt 0 l 3 3 τ ≤N 3 τ =1,...,t
≪ c ′′ (α, ϵ)(log N )s+t +ϵ . Lemma 5. (a) For almost all α for all ϵ > 0 there is a c (α, ϵ) such that for all V ∈ N, all v ≥ −V , all N and all l ∈ Nt0 we have
2v # n ∈ l-box ∥nb(j )α∥ < 2 2 ¯ n¯ 1 · · · logn ¯ t ¯l1 · · · ¯lt n¯ 1 · · · n¯ t log
j
≤ c (α, ϵ) · 2v V max(1, v)1+ϵ (log N )s+ϵ . (b) For almost all α for all ϵ > 0 there is a c˜ (α, ϵ) such that for all N and all v < −˜c (α, ϵ) log log N, all l ∈ Nt0 and for all j we have
2v # n ∈ l-box ∥nb(j )α∥ < 2 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯lt n¯ 1 · · · n¯ t logn
= 0.
Proof of Lemma 5. (a) For ϵ > 0, V > 0 and v ≥ −V consider v
[0,1)t w=−V
1 2w V
max(1, w)
1+ϵ
(¯j1 · · · ¯js )−1−ϵ l ∈Nt0
j
2w dα × # n ∈ l-box ∥nb(j )α∥ < 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯l2t n¯ 1 · · · n¯ t logn v 1 −1−ϵ ¯ ¯ = (j1 · · · js ) 1dα, 2w V max(1, w)1+ϵ j t n∈l-box A w=−V
l ∈N0
where A :=
α ∈ Rt ∥nb(j )α∥ <
2w
¯ 1 · · · logn ¯ t ¯l21 · · · ¯l2t n¯ 1 · · · n¯ t logn
.
Since
1dα = A
2w
¯ 1 · · · logn ¯ t ¯l21 · · · ¯l2t n¯ 1 · · · n¯ t logn
the last expression is
≤
v
1 V max(1, w)
1+ϵ
w=−V
(¯j1 · · · ¯js )−1−ϵ j
l ∈Nt0
1
¯l21 · · · ¯l2t
1
n∈l-box
¯ 1 · · · logn ¯ t n¯ 1 · · · n¯ t logn
≪ c (ϵ).
¯ 1 · · · logn ¯ t )−1 ≪ 1.) (Note that n∈l-box (¯n1 · · · n¯ t logn Therefore for almost all α for all ϵ > 0 there is a c (α, ϵ) such that for all v ≥ −V and all N we have
l ∈Nt0
j
2v # n ∈ l-box ∥nb(j )α∥ < 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯l2t n¯ 1 · · · n¯ t logn (¯j1 · · · ¯js )1+ϵ 1
≤ c (α, ϵ)2v V max(1, v)1+ϵ (log N )s+ϵ . From this part (a) of Lemma 5 immediately follows.
G. Larcher / Journal of Complexity 29 (2013) 397–423
411
(b) Let α be such that part (a) of Lemma 5 is satisfied, and let c (α, ϵ) be the constant from part (a) of Lemma 5 and c˜ (α, ϵ) be such that c (α, ϵ)2−V V (log N )s+ϵ < 1 for V > c˜ (α, ϵ) log log N. Then for all such V and v = −V by part (a) we have
2v =0 # n ∈ l-box ∥nb(j )α∥ < 2 ¯ 1 · · · logn ¯ t ¯l1 · · · ¯l2t n¯ 1 · · · n¯ t logn
for all l and j, and part (b) follows.
Lemma 6, finally, is a generalization of Lemma 4.2 in [2]. Lemma 6. For j given, let Vj = (vj (1), . . . , vj (t )), Wj = (wj (1), . . . , wj (t )) be such that 0 ≤ vj (τ ) < wj (τ ) for τ = 1, . . . , t. For n = (n1 , . . . , nt ) ∈ Nt0 we write Vj ≤ n < Wj if vj (τ ) ≤ nτ < wj (τ ) for τ = 1, . . . , t. For arbitrary cj = q4j with some qj ∈ N let
Z (α, j , cj , Vj , Wj ) := # n Vj ≤ n < Wj with {nb(j )α } <
cj
n¯ 1 · · · n¯ t
and cj
E (j , cj , Vj , Wj ) :=
Vj ≤n
n¯ 1 · · · n¯ t
.
Let J ⊆ {j = (j1 , . . . , jt ) | 0 ≤ ji < Li for i = 1, . . . , s} . Then for almost all α for all ϵ > 0 there is a c (α, ϵ) such that for all J , Vj , Wj and qj we have
Z (α, j , cj , Vj , Wj ) =
j ∈J
E (j , cj , Vj , Wj ) + E
j ∈J
with
|E | ≤ c (α, ϵ)
3
1
ϵ
cj4 (log cj )t (¯j1 · · · ¯js ) 2 + 2 (log N )
1 +ϵ t 2
.
j ∈J
Proof of Lemma 6. We proceed quite analogously (with adapted details) as Beck in the proof of Lemma 4.2 in [2]. For some parts of the proof of our Lemma 6 we will directly refer to the proof of Lemma 4.2. in [2]. So we will also refer for some of the necessary definitions (which are used in this proof only) to the proof of 4.2 in [2], and we do not give them all here in this paper again. By Lemma 9.3. in [2] for every j we have
[0,1)t
(Z (α, j , cj , Vj , Wj ) − E (j , cj , Vj , Wj ))2 dα = 4
Vj ≤n
div(n)
cj n¯ 1 · · · n¯ t
,
where div(n) = div(n1 , . . . , nt ) denotes the number of common divisors of n1 , . . . , nt . In the same way as in the proof of Lemma 4.2 in [2, pp. 496 and 497], from this we obtain (for the definitions of ‘special boxes’, ‘size’, ‘order’ see there):
C14
[0,1)t
˜ ) − E (j , cj , V˜ , W ˜ ))2 dα ≪ htj · cj · (2p )t . (Z (α, j , cj , V˜ , W
(16)
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G. Larcher / Journal of Complexity 29 (2013) 397–423
Here condition C14 means C14 : ˜ ) in the sense of Lemma 4.2 in [2] of size
p
The integer p is defined such that e2 ≤ N < e2 . For κ > 0 and ϵ > 0 let (note that cj = q4j for some qj = 1, 2, . . .)
˜ ) − E (j , cj , V˜ , W ˜ ))2 Bκ (p, hj , j , qj ) := α (Z (α, j , cj , V˜ , W C 14 ≥ κ · htj · q6j · (2p )t · p2 · (¯j1 · · · ¯js )1+ϵ . Then for the measure of Bκ from (16) it follows that
λ(Bκ (p, hj , j , qj )) ≪
1
κ · q2j · p2 · (¯j1 · · · ¯js )1+ϵ
and therefore
λ(Bκ (p, hj , j , qj )) ≤
j ,p,qj
1
κ
c (ϵ).
With Bκ :=
j ∈J p,qj
Bκ (p, hj , j , qj ) c (ϵ)
then we have λ(Bκ ) ≤ δ for κ ≥ δ and for every J. Let now hj := pt + c¯ log q4j for a certain absolute constant c¯ such that q4i · 2tp −hj ≪ 1. Let ϵ > 0, then for almost all α ∈ Rt there is a κ(ϵ, t ) such that α ̸∈ Bκ for all κ > κ(ϵ, t ). That means, for almost all α ∈ Rt there is a c (α, ϵ, t ) such that for all j , p, qj (with hj chosen like above) we have
˜ ) − E (j , cj , V˜ , W ˜ ))2 ≪ c (α, ϵ, t ) · htj · q6j · (2p )t · p2 · (¯j1 · · · ¯js )1+ϵ . (Z (α, j , cj , V˜ , W C14
From this, exactly in the same way as it is carried out in the proof of Lemma 4.2. in [2] on pp. 498 and 499 we obtain: (note that there are typing errors in formulas (9.66) and (9.67) in [2], the correct form of (9.66) is
2
˜ ) − E (c ; V˜ , W ˜ ) ≪ · · · Z (α; c , V˜ , W
˜) some special boxes Q (V˜ ,W
and the correct form of (9.67) is
Z α; c , upper lower
bound
−E
upper lower
≪ · · · .) bound
For α and ϵ > 0 from above, we have for all j , p, qj that
1 ϵ t ˜ ˜ ˜ ˜ Z (α, j , cj , V , W ) − E (j , cj , V , W ) ≪ c (α, ϵ, t ) · htj · q3j · (2p ) 2 · p · (¯j1 · · · ¯js ) 2 + 2 . C 14
G. Larcher / Journal of Complexity 29 (2013) 397–423
413
Hence (again using the arguments of Beck on pp. 498 and 499) for almost all α
Z (α, j , cj , Vj , Wj ) =
j ∈J
E (j , cj , Vj , Wj ) + E
j ∈J
with (note that hj is chosen such that q4j · 2tp−hj ≪ 1 and that p ≪ log log N)
|E | ≪
1
t
ϵ
c (α, ϵ, t ) · htj · q3j · (2p ) 2 · p · (¯j1 · · · ¯js ) 2 + 2 + q4j · 2tp−hj
j ∈J
≪ c (α, ϵ)
3
1
ϵ
cj4 (log cj )t (¯j1 · · · ¯js ) 2 + 2 (log N )
1 +ϵ t 2
j ∈J
for all N , J , Vj , Wj , qj = 1, 2, . . . and cj = q4j .
4. The proof of the theorem For the proof of the theorem we can restrict to t ≥ 2, since it already was shown for t = 1 in [9]. t By the considerations in Section 2 it remains to show that for almost all α ∈ R we have that the (for almost all α) absolutely convergent series satisfies for all ϵ > 0:
:=
j
n∈Zt +1 \{0}
pro(N , x, θ, j , n) ≤ c (α, ϵ, b1 , . . . , bs )
(log N )s+t +ϵ N
for arbitrary x = (x1 , . . . , xt ) and arbitrary θ = θ(j ) = (θ1 (j ), . . . , θt (j )), where pro(N , x, θ, j , n) := i
t −1
t 1 − e2π inτ xτ 2π nτ
τ =1
· g (n, j ) ·
t
1 − e2π i(b(j )(n1 α1 +···+nτ ατ )−nt +1 )N (j ) 2π (b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )
e2π iθτ (j ) ,
τ =1
where
2 2 t sin 2π Nn(jτ)2 sin(2π (b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )) g (n, j ) := · . 2π Nn(jτ)2 2π (b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 ) τ =1 2π inτ xτ = −ixτ and Here we always – without extra noting it – assume 1−e2π n τ
nτ N (j )2 nτ N (j )2
sin 2π 2π
= 1 if nτ = 0.
Note that always
|pro(N , x, θ, j , n)| 2 −1 nτ · max 1, ≤ · n¯ 1 · · · n¯ t |(b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )| N (j )2 τ =1 1
≤
1
1 n¯ 1 · · · n¯ t
·
1
|(b(j )(n1 α1 + · · · + nτ ατ ) − nt +1 )|
≤
t
1
1
n¯ 1 · · · n¯ t ∥nb(j )α∥
holds. If in the following we write LHS ≪ RHS, then we always mean that LHS ≤ c · RHS, with a constant c depending at most on t and on s (and not on N , x, α, . . .). We now proceed essentially along the same lines as Beck did in [2], by dividing in various parts, but with the necessary adaptations. Let first
1
:=
j
n∈Zt \{0} nt +1 ∈Z C1
pro(N ; x, θ, j , n),
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G. Larcher / Journal of Complexity 29 (2013) 397–423
where condition C1 means: C1 : n = (n1 , . . . , nt ) with max |nτ | > N (j )2 (log N (j ))t . 1≤τ ≤t
Then
≪
1
2
+
, 3
where
2 −1 nτ , · max 1, N (j )2 τ =1
:=
j
2
(|¯n1 · · · n¯ t | · ∥nb(j )α∥)
−1
n∈Zt \{0} C1
t
and
:=
j
3
∞
t
τ =1
r =1 n∈Zt \{0}
2 −1 nτ 1 |nτ | max 1, · 3. 2 N (j ) r
C1
Now (to obtain the following first inequality set max |nτ | := κ · N (j )2 · (log N (j ))t with κ ≥ 1 in the above expression and the result will follow)
≪
max(1, |nτ |(log |nτ |)2 )
j
3
≪
2
n∈Zt \{0} τ =1
∞ j
For
∞ −1 1
t
n =2
r =1
t
1 n(log n)2
·
∞ 1 r =1
r3
r3
≪ (log N )s .
we obtain in the same way as above (note that we can restrict to t ≥ 2):
≪
j
2
τ =1
n∈Zt \{0}
≪
j
−1 t 2t ¯ 1 · · · nt ) ∥nb(j )α∥ max(1, logn n¯ τ
t ¯ 1 · · · nt ) t + 2 ∥nb(j )α∥ max(1, logn n¯ τ
−1 ,
τ =1
n∈Zt \{0}
hence by Lemma 3 it follows, that for almost all α for every ϵ > 0 there is a c (α, ϵ) such that
≪ c (α, ϵ)(log N )s+ϵ .
2
Next we handle
4
:=
j
pro(N ; x, θ, j , n),
n∈Zt \{0} C2 ,C3
where condition C2 means: C2 : n = (n1 , . . . , nt ) with max |nτ | ≤ N (j )2 (log N (j ))t 1≤τ ≤t
and where condition C3 means: C3 :
|b(j )(n1 α1 + · · · + nt αt ) − nt +1 | ≥
1 3
.
G. Larcher / Journal of Complexity 29 (2013) 397–423
415
The inner sum n in 4 is estimated for every α in exactly the same way as the sum 4 in [2, Section 5], to be ≪ (log N )t (notethat in [2] the letter k is our parameter t), with an implied constant independent of j. Hence our 4 satisfies
≪ (log N )s+t
4
for every α. Next let
:=
j
5
pro(N ; x, θ, j , n),
n∈Zt \{0} nt +1 ∈Z C5 C2 ,C4
where condition C4 means: C4 :
|b(j )(n1 α1 + · · · + nt αt ) − nt +1 | ≤
(log N )20(t +s) |¯n1 · · · n¯ t |
and where condition C5 means: C5 : nt +1 is such that |b(j )(n1 α1 + · · · nt αt ) − nt +1 | = ∥nb(j )α∥ . Then by Lemma 4 for almost all α there is for all ϵ > 0 a c (α, ϵ) such that
≤
j
5
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1 ≤ c (α, ϵ)(log N )s+t +ϵ .
n∈Zt \{0} C2 , C4
In the next step we handle two parts of the following sum
:=
j
6
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1
n∈Zt \{0} C6 ,C7
where condition C6 means: C6 :
|nτ | ≤ N (j )2 (log N (j ))t for τ = 1, . . . , t , and where condition C7 means: C7 : 1 3
> ∥nb(j )α∥ >
The two parts
(j )
71
(log N )20(t +s) . |¯n1 · · · n¯ t |
and (j )
72
of
6
which we treat first are the following:
let vj = (v1 , . . . , vt ) with (j ) 1 ≤ 2vτ ≤ N (j )2 (log N (j ))t
for all τ = 1, . . . , t ,
and (j ) (j ) T (vj ) := n ∈ Zt \ {0} | 2vτ −1 ≤ n¯ τ < 2vτ , 1 ≤ τ ≤ t ,
then
71
:=
j
vj
n∈T (vj )
(j ) min vτ ≪log log N τ
C6 ,C7
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1
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G. Larcher / Journal of Complexity 29 (2013) 397–423
and
:=
72
j
vj
n∈T (vj )
(j ) N (j )2 max 2vτ ≥ 4 τ
(∥nb(j )α∥ n¯ 1 · · · n¯ t )−1 .
C6 ,C7
Let us consider the first part:
≪
71
j
vj
1
(j ) min vτ ≪log log N τ
q(j ) C8
2q(j )
#T (vj , q(j ), j )
where condition C8 means: C8 :
(log N )20(t +s) ≤ 2q(j ) ≤ 2N 2t (log N )20(t +s)t , and where T (vj , q(j ), j ) := n ∈ T (vj )|2q(j )−1 < n¯ 1 · · · n¯ t ∥nb(j )α∥ ≤ 2q(j ) .
(Note, that n¯ 1 · · · n¯ t ∥nb(j )α∥ ≤ N (j )2t (log N (j ))t
2
always trivially holds, since nτ < N (j )2 (log N (j ))t for all τ .) Hence
71
1
≪
v =(v1 ,...,vt ) min vτ ≪log log N
τ
q C8
2q
#T (v , q, j ).
j
By Lemma 6, for almost all α for all ϵ > 0 there is a c (α, ϵ) such that
# T (v , q, j ) ≪
j
2q−1
j
(j) (j) 2vτ −1 ≤¯nτ <2vτ
+ c (α, ϵ)
n¯ 1 · · · n¯ t
3q 1 ϵ 2 4 +ϵ (¯j1 · · · ¯js ) 2 + 2 (log N )
j
≪ c (α, ϵ)2q (log N )s since 2q ≥ (log N )20(t +s) . Hence
71
≪ c (α, ϵ)
v =(v1 ,...,vt ) min vτ ≪log log N
τ
(log N )s q C8
≪ c (α, ϵ)(log N )t −1 log log N · log N · (log N )s ≪ c (α, ϵ)(log N )t +s log log N . The second part is handled quite analogously and leads to
72
≪ c (α, ϵ) ·
v =(v1 ,...,vt ) 2 N (j)2 N (j)2 (log N (j))t >max 2vτ > 4 τ
(log N )s q C8
≪ c (α, ϵ)(log N )t −1 log log N · log N · (log N )s ≪ c (α, ϵ)(log N )t +s log log N .
1 +ϵ t 2
G. Larcher / Journal of Complexity 29 (2013) 397–423
417
So it remains to estimate the following sum (i.e., the analogue to the work done by Beck in his Sections 6 and 7 in [2]):
:=
pro(N ; x, θ, j , n),
n∈Zt \{0} C7 ,C9
j
8
where condition C9 means: C9 :
(log N )20(t +s) ≤ min |nτ | ≤ max |nτ | ≤
N (j )2 4
.
(Note that condition C7 implies C5 .) It is not sufficient to handle 8 by just estimating absolute values of the summands. So in analogy to Beck [2, p. 469], we write (here and in the following an nτ in the denominator which equals 0 is replaced by 1)
=
8
i t −1
(2π )t +1
Λ(θ, j )
n∈Zt \{0} C7 ,C9
j
n1 · · · nt (nb(j )α − nt −1 )
δ1 +···+δt +1
×
g (n, j )
Λ(θ, j )
j
e2π iL1,j (n,α)
(−1)
n1 · · · nt (nb(j )α − nt +1 )
n∈Zt \{0} C7 , C9
1=(δ1 ,...,δt +1 )∈{0,1}t +1
+
where t
Λ(θ, j ) :=
e2π iθτ (j ) ,
τ =1
and L1,j (n, α) := δt +1 N (j )(nb(j )α − nt +1 ) +
t
δτ xτ nτ .
τ =1
We start the analysis with the first part of
:=
9
Λ(θ, j )
8,
i.e., with
g (n, j )
n∈Zt \{0} C7 ,C9
j
n1 · · · nt (nb(j )α − nt −1 )
.
For 1
, (log N )2 l = (l1 , . . . , lt +1 ) ∈ Nt0+1 δ :=
and
ϵ = (ϵ1 , . . . , ϵt +1 ) ∈ {−1, 1}t +1 let
U (l , ϵ, j ) := n ∈ Zt | (1 + δ)lτ ≤ ϵτ nτ < (1 + δ)lτ +1
for 1 ≤ τ < t ,
and
(1 + δ)lt +1 ≤ ϵτ +1 n1 · · · nτ (nb(j )α − nt +1 ) < (1 + δ)lt +1 +1 .
g (n, j ) ,
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G. Larcher / Journal of Complexity 29 (2013) 397–423
By C9 for given j the number of sets U (l , ϵ, j ) which are relevant for
log N t +1
8
(and therefore for
. at most ≪ δ For any choice of
ϵ˜ := (ϵ1 , . . . , ϵt ) ∈ {−1, 1}t let
ϵ˜ + := (ϵ1 , . . . , ϵt , 1) ∈ {−1, 1}t +1 and
ϵ˜ − := (ϵ1 , . . . , ϵt , −1) ∈ {−1, 1}t +1 . Then
=
Λ(θ, j )
j
9
l
ϵ˜ ∈{−1,1}t
n∈U (l ,˜ϵ+ ,j )
n1 · · · nt (nb(j )α − nt −1 )
g (n, j )
+
g ( n, j )
n∈U (l ,˜ϵ− ,j )
n1 · · · nt (nb(j )α − nt −1 )
=
l ,˜ϵ
Λ(θ, j )
+
n∈U (l ,˜ϵ+ ,j )
j
.
n∈U (l ,˜ϵ− ,j )
We will need now the following technical lemma
˜ ∈ U (l , ϵ˜ , j ) satisfying C7 and C9 , then Lemma 7. Let n ∈ U (l , ϵ˜ , j ) satisfying C7 and C9 and n +
−
˜ , j) g (n, j ) δ g (n + n · · · n (nb(j )α − n ) n˜ · · · n˜ (nb(j )α − n˜ ) ≪ (1 + δ)lt +1 . 1 t t +1 1 t t +1 Proof of Lemma 7. Note that 0 < nb(j )α − nt +1 <
1 3
and
−
1 3
˜ (j )α − n˜ t +1 < 0. < nb
Further 1 3
> |nb(j )α − nt +1 | ≥
(1 + δ)lt +1 = (1 + δ)lt +1 −(l1 +···lt )−t (1 + δ) · · · (1 + δ)lt +1 l1 +1
hence lt +1 ≤ (l1 + · · · + lt ) + t . We want to show
˜ , j) g (n, j ) g (n δ ≪ − . n1 · · · nt |nb(j )α − nt +1 | n˜ 1 · · · n˜ t nb(j )α − n˜ t +1 (1 + δ)lt +1 If we can show, that
g (n, j ) − g (n˜ , j ) ≪ δ,
9)
is
G. Larcher / Journal of Complexity 29 (2013) 397–423
419
then we have
˜ , j) + ϵ g (n, j ) = g (n for some ϵ with |ϵ| ≪ δ , hence:
˜ , j) g (n g (n, j ) − n1 · · · nt |nb(j )α − nt +1 | n˜ 1 · · · n˜ t nb ˜ (j )α − n˜ t +1 ˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb 1 (g (n˜ , j ) + ϵ) − g (n˜ , j ) , = n1 · · · nt |nb(j )α − nt +1 | ˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb and
˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb − g (n˜ , j ) (g (n˜ , j ) + ϵ) n1 · · · nt |nb(j )α − nt +1 | ˜ (j )α − n˜ t +1 ˜ (j )α − n˜ t +1 n˜ 1 · · · n˜ t nb n˜ 1 · · · n˜ t nb = g (n˜ , j ) −1 +ϵ n1 · · · nt |nb(j )α − nt +1 | n1 · · · nt |nb(j )α − nt +1 | 1 ≤ max ((1 + δ) − 1) , 1 − + δ(1 + δ) ≪ δ, (1 + δ) hence
˜ , j) δ g (n, j ) g (n ≪ − . n1 · · · nt |nb(j )α − nt +1 | n˜ 1 · · · n˜ t nb(j )α − n˜ t +1 (1 + δ)lt +1 It remains to show that
g (n, j ) − g (n˜ , j ) ≪ δ. Note that N2π2 (njτ) ≤ π2 and |2π (nb(j )α − nt +1 )| ≤ Using show
sin x x
= 1−
x2
+ ϵ(x)x with |ϵ(x)| ≤ 4
6
2π 3
1 120
.
for − 23π ≤ x < 23π , it now is an easy exercise to
g (n, j ) − g (n˜ , j ) ≪ (1 + δ)lt +1 −l1 −···−lt δ ≪ δ, since lt +1 ≤ (l1 + · · · + lt ) + t . The proof of Lemma 7 is finished.
We proceed now with the proof of the theorem, i.e., first with the estimation of For l , ϵ˜ given, we consider
˜ := Λ(θ, j )
n∈U (l ,˜ϵ+ ,j )
j
+
n∈U (l ,˜ϵ− ,j )
Let U :=
j
g (n, j )
U (l , j , ϵ˜ ). +
n1 · · · nt (nb(j )α − nt −1 )
g (n, j ) n1 · · · nt (nb(j )α − nt −1 )
.
9:
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G. Larcher / Journal of Complexity 29 (2013) 397–423
By Lemma 6 we have for almost all α for all ϵ > 0, with C1 := (1 + δ)lt +1 +1 and C2 := (1 + δ)lt +1 : #U =
C1 − C2
n∈Zt \{0}
j
+ R ≤ (C1 − C2 )δ t (log N )s + R
n¯ 1 · · · n¯ t
(1+δ)lτ ≤nτ <(1+δ)lτ +1
with 3 1 1 +ϵ t 4 +ϵ + 2ϵ 2 ¯ ¯ 2 |R| ≪ c (α, ϵ) (j1 · · · js ) (log N ) C1 . j
g (n,j )
Hence (note that |Λ(θ, j )| ≤ 1 and n ···n (nb(j )α−n ) ≤ t 1 t −1
1 ), C1
altogether we have:
˜ ≤ (C − C )δ t (log N )s δ + c (α, ϵ) (¯j · · · ¯j ) 12 + 2ϵ (log N ) 21 +ϵ t C 34 +ϵ · 1 1 s 1 2 1
C1
C1
j 3
1
≪ δ t +2 (log N )s + c (α, ϵ)(log N ) 2 s+ 2 ϵ s (log N )
1 +ϵ t 2
1
·
1 −ϵ 4
.
C1 Hence (note that by condition C7 we have C1 > (log N )20(t +s) )
1 (log N )t +1 t +2 3 +ϵ t s s+ 12 ϵ s 2 2 δ ( log N ) + c (α, ϵ)( log N ) ( log N ) · ≤ 9 δ t +1 ≪ c (α, ϵ)(log N )
t +s−1
1 1 −ϵ 4
C1
.
To finish the proof of the theorem it remains to estimate
:=
e2π iL1,j (n,α)
n∈Zt \{0} C7 ,C9
1=(δ1 ,...,δt +1 )∈{0,1}t +1
j
10
(−1)δ1 +···+δt +1
Λ(θ, j )
n1 · · · nt (nb(j )α − nt +1 )
g (n, j ).
For estimating 10 we need a generalization of Beck’s ‘Key Lemma’ provided in [2]. We define in analogy to [2, Section 7]: Definition 1. For given 1, j and l = (l1 , . . . , lt +1 ) ∈ Nt0+1 satisfying
(log N )20(t +s) ≤ 1 +
lτ
1
(log N )
2
<
N (j )2
for τ = 1, . . . , t
4
(17)
and 20(t +s)
(log N )
≤ 1+
1
lτ +1
(log N )2
<
N (j )2 4
t
,
(18)
we say that l is an ϵ-big-vector for b(j )α if
± U (l , j , ϵ) 2π iL1,j (n,α) 2π iL1,j (n,α) e − e , ≥ log N n∈U (l ,j ,ϵ+ ) n∈U (l ,j ,ϵ− ) where U ± (l , j , ϵ) := U (l , j , ϵ+ ) ∪ U (l , j , ϵ− ). Further we define the concept of a ‘special ϵ -line’ in the same way as it is done in [2], Definitions 7.2 and 7.3. For the sake of completeness we repeat the definitions here:
G. Larcher / Journal of Complexity 29 (2013) 397–423
421
Definition 2. Two integral vectors l = (l1 , . . . , lt +1 ) and h := (h1 , . . . , ht +1 ) satisfying the conditions as stated in (17) and (18) for l, are called ϵ-neighbors if
(1 + δ)ϵτ (hτ −lτ ) = (log N )9 for 1 ≤ τ ≤ t and, (1 + δ)(ht +1 −lt +1 ) = (log N )9(t +1) . ϵ
The notation l → h means that the ordered pair ⟨l , h⟩ of vectors satisfies the two conditions above. Definition 3. A sequence H = h(1) , h(2) , . . . of vectors satisfying the conditions as stated in (17) and (18) for l, is called a ‘special ϵ-line’ if
ϵ ϵ ϵ h(1) → h(2) → h(3) → . . . ,
that is, any two consecutive vectors in H are ϵ-neighbors. We will show a generalized form of Beck’s ‘Key Lemma’, namely. Lemma 8. For every α satisfying Lemma 6, (i.e., for almost all α) we have: For given 1 there is an N (α) such that for all N > N (α) and all j we have that every special ϵ-line contains at most one ϵ-big vector for b(j )α. Remark 3. It seems that from the Key Lemma of Beck as it is stated in [2], our generalized version of the Key Lemma, i.e., Lemma 8, follows immediately for almost all α. However, the formulation of Beck’s Key Lemma as it is stated in [2] is not quite complete. The complete formulation would be the following: ‘‘For any α ∈ Rt satisfying Lemma 4.2 (in [2]) there is an N (α), such that for all N ≥ N (α) every special ϵ-line contains at most one ϵ-big vector.’’ From this correct version our Lemma 8 does not follow immediately. Proof of Lemma 8. In the proof we can follow the proof of Beck’s Key Lemma almost line by line with the only two exceptions that we have to use our Lemma 6 (in suitable form) wherever (essentially two times) Beck uses his Lemma 4.2. We start with the proof of Lemma 8 until we have to use Lemma 6 for the first time. Further we show where to use Lemma 6 for the second time. The rest of the proof can be carried out quite analogously as it is done in the proof of the Key Lemma by Beck in [2]. So, for given 1 and j let H = h(1) , h(2) , . . . be a special ϵ-line with two ϵ-big vectors h(p) and h(q) for b(j )α. If
L1,j (n, α) ≤
1
for every n ∈ U ± (h(p) , j , ϵ)
(log N )2
(19)
then
1 − e2π iL1,j (n,α) ≤
1
(log N )2
for all n ∈ U ± (h(p) , j , ϵ)
(20)
and hence
2π iL1,j (n,α) 2π iL1,j (n,α) e − e n∈U (h(p) ,j ,ϵ+ ) n∈U (h(p) ,j ,ϵ− ) ± (p) U (h , j , ϵ) (p) + (p) − ≪ U (h , j , ϵ ) − U (h , j , ϵ ) + , (log N )2 and this by Lemma 6 with J = {j } is at most
± (p) 3+ 1 U (h , j , ϵ) 1 1 (p) 1 1 + 10 t ht +1 4 10 + 2 ¯ ¯ . ≪ c (α) (1 + δ) (log N ) (j1 · · · js ) 2 10 + (log N )2
(21)
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G. Larcher / Journal of Complexity 29 (2013) 397–423
Further, by Lemma 6 and again with δ = (log1N )2 , C1 = (1 + δ)lt +1 +1 and C2 = (1 + δ)lt +1 we have
± (p) U (h , j , ϵ) ≫ (C1 − C2 )
1
(log N )2t
34 + 101 1 (p) 1 1 +1 t − c (α) (1 + δ)ht +1 (log N ) 2 10 (¯j1 · · · ¯js ) 2 + 10 34 + 101 1 (p) (p) 1 1 +1 t ≫ δ(1 + δ)ht +1 δ t − c (α) (1 + δ)ht +1 (log N ) 2 10 (¯j1 · · · ¯js ) 2 + 10 (p)
≫ δ t +1 (1 + δ)ht +1 for N ≥ N (α) (p) ht +1
≥ (log N )20(s+t ) . (p) h Therefore (again also since (1 + δ) t +1 ≥ (log N )20(s+t ) ) we have ± (p) U (h , j , ϵ) for N ≥ N (α). (21) ≪ (log N )2
since (1 + δ)
But this contradicts the assumption that h(p) is an ϵ-big vector for b(j )α. From here on we now proceed quite analogously as in the proof of the Key Lemma in [2], starting with formula (7.8) in Beck’s proof: The falsity of the assumption in (19) means, that for every α satisfying Lemma 6 there is an N (α) such that for all j we have that there is an n∗ ∈ U ± (h(p) , j , ϵ) such that
L1,j (n∗ , α) >
1
(log N )2
for all N ≥ N (α). In formula (7.23) Beck uses his Lemma 4.2 to estimate the maximal number of border points. At this place we again use our Lemma 6 with an implied constant c (α) not depending on j, quite analogously as we did at the beginning of this proof. This gives the proof of Lemma 8. Now we are ready to estimate We set
˜ (j , 1) := 10
n∈Zt \{0} C7 ,C9
10
and to finish the proof of the theorem:
e2π iL1,j (n,α) n1 · · · nt (nb(j )α − nt +1 )
g (n, j ).
This is just the sum 10 in the paper of Beck in [2] now with b(j )α instead of α. Beck’s 10 is estimated in the first part of Section 8 in [2] to be ≤ c (α)(log N )t log log N for all α satisfying Beck’s Lemma 4.2, by using Beck’s Key Lemma and Beck’s Lemma 4.2 (note that in [2] the symbol k stands for our symbol t). Quite analogously, by using our generalized Key Lemma (Lemma 8) instead of Beck’s Key Lemma, and using our Lemma 6 instead of Beck’s Lemma 4.2 we get
˜ 10 (j , 1) ≤ c (α)(log N )t log log N for all α satisfying Lemma 6 and for all j and with C (α) not depending on j.
(By using Beck’s auxiliary results we just would obtain ˜
10
(j , 1) ≤ c (α, j )(log N )t log log N,
i.e., an estimate with a constant c (α, j ) depending also on j.) Now of course, for all α satisfying Lemma 6, i.e., for almost all α, we have
≤ c (α)(log N )s+t log log N . 10 This finishes the proof of the theorem.
G. Larcher / Journal of Complexity 29 (2013) 397–423
423
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