- Email: [email protected]
On convergence almost everywhere of series of dilated functions
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
[m3G; v1.159; Prn:26/08/2015; 8:57] P.1 (1-4)
C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
Contents lists available at ScienceDirect
C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com
Number theory/Harmonic analysis
On convergence almost everywhere of series of dilated functions Sur la convergence presque partout des séries de fonctions dilatées Michel J.G. Weber IRMA, 10, rue du Général-Zimmer, 67084 Strasbourg cedex, France
a r t i c l e
i n f o
Article history: Received 28 July 2014 Accepted after revision 27 July 2015 Available online xxxx Presented by Jean-Pierre Kahane
a b s t r a c t
2 2iπ x Let f (x) = , where ∈Z a e k≥1 ak d(k) < ∞ and d(k) = d|k 1 and let f n (x) = f (nx). We show by using a new decomposition of squared sums that, for any K ⊂ N ∞ ∞ sin 2π jx 2 finite, k∈ K ck f k 22 ≤ ( m=1 am d(m)) k∈ K ck2 d(k2 ). If f s (x) = , s > 1/2, by j =1 js only using elementary Dirichlet convolution calculus, we show that for 0 < ε ≤ 2s − 1, ζ (2s)−1 k∈ K ck f ks 22 ≤ 1+ε ε ( k∈ K |ck |2 σ1+ε−2s (k)), where σh (n) = d|n dh . From this, we
deduce that if f ∈ BV(T), f , 1 = 0 and
∞
4 2 (log log k) k=1 ck (log log log k)2
< ∞, then the series
k ck f k
converges almost everywhere. This slightly improves a recent result, depending on a fine ∞ analysis on the polydisc ([1], th. 3) (nk = k), where it was assumed that k=1 ck2 (log log k)γ converges for some γ > 4. © 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
2 Soit f (x) = ∈Z a e2iπ x telle que la série k≥1 ak d(k) où d(k) = d|k 1 converge, et soit f n ( x) = f ( nx ) . Nous montrons à l’aide d’une nouvelle décomposition des sommes carrées ∞ 2 que k∈ K ck f k 22 ≤ ( m=1 am d(m)) k∈ K ck2 d(k2 ), pour tout ensemble fini d’entiers K .
Si f s (x) =
∞
j =1
sin 2π jx , js
s > 1/2, nous montrons aussi, par un calcul simple sur les
ε ≤ 2s − 1 et σh (n) = f , 1 = 0, si la série
∞
ε 2 ck f ks 22 ≤ 1+ ε ( k∈ K |ck | σ1+ε−2s (k)), où 0 < h d . Nous en déduisons que, pour tout f ∈ BV(T) telle que d|n
convolutions de Dirichlet, que ζ (2s)−1
4 2 (log log k) k=1 ck (log log log k)2
k∈ K
converge, alors la série
k ck f k
converge presque
partout. Cela améliore un résultat récent, dépendant analyse fine sur le polydisque ∞ d’une 2 γ ([1], th. 3) (nk = k), où l’on suppose que la série k=1 ck (log log k) converge pour un réel γ > 4. © 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.crma.2015.07.010 1631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
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M.J.G. Weber / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
2
1. Introduction – main result One of the oldest and most central problems ∞ in the theory of systems of dilated sums is the study of the convergence in norm or almost everywhere of the series k=1 ck f (nk x), where f is a periodic function f and N = {nk , k ≥ 1} a sequence of positive integers (see [3]). Our main concern is the search of individual conditions ensuring convergence, a barely investigated part of the theory. We use an arithmetical approach based on elementary Dirichlet convolution calculus and on a new decomposition of squared sums, continuing the work in [7,4]. We show that this approach is strong enough to recover and even slightly improve a recent a.e. convergence result [1] (Theorem 3) in the case N = N without using analysis on the polydisc. Results in [1] were recently developed in [2]. Our approach is also in the spirit of the work of Hilberdink [5] on some arithmetical mappings and extrema linked to arithmetical functions, with applications ∞ to -results of the Riemann Zeta function. Denote e (x) = e2iπ x , en (x) = e (nx), n ≥ 1. Let T = R/Z = [0, 1[. Let f (x) ∼ j =1 a j e j (x). Let f n (x) = f (nx), n ∈ N. We assume throughout that
f ∈ L 2 (T),
f , 1 = 0.
(1)
A key preliminary step naturally consists in searching bounds of k∈ K ck f k 2 integrating in their formulation the arithmetical structure of K . That question has received a satisfactory answer only for specific cases. We state our mean results. Let d(n) be the divisor function, namely the number of divisors of n. Throughout, K denotes a finite set of natural numbers. Theorem 1.1. Assume that
∞
2 m=1 am d(m)
< ∞. Then,
∞ 2 2 2 ck f k ≤ a2 d(m) c d(k ). 2
k∈ K
m
k
m =1
k∈ K
In [7], using Hooley’s Delta function, we recently showed a similar estimate however restricted to sets K such that r r +1 K ⊂]e , e ] for some integer r. Theorem 1.1 is deduced from a more general result. Introduce the necessary notation. Let ∞ A k = ν =1 a2ν k . Let ζh be defined by ζh (n) = nh for all positive n. Let θ(n) denotes the number of squarefree divisors of n. Given K ⊂ N, we note F ( K ) = {d ≥ 1; ∃k ∈ K : d|k}. If K is factor closed (d|k ⇒ d ∈ K for all k ∈ K ), then F ( K ) = K . Theorem 1.2. Let ψ be any arithmetical function taking only positive values. Then,
2 ck f k 2 ≤ B ck2 ψ ∗ ζ0 (k), k∈ K
where
B = sup
A dk
d∈ F ( K )
k∈ K
k∈ K d|k
ψ( dk )
k θ( ) < ∞. d
Here ∗ denotes the Dirichlet convolution. By choosing ψ = θ and since ψ ∗ ζ0 (k) = d(k2 ), we check that B ≤ ∞ sin 2π jx 2 s , s > 1/2 m=1 am d(m), whence Theorem 1.1. Consider now the class of functions introduced in [6], f (x) = j =1 js
∞
2s
and recall that f ks , f s = ζ (2s) (kk,) s s
where (k, ) = gcd(k, ).
Theorem 1.3. Let s > 0, 0 ≤ τ ≤ 2s. Let also ψ1 (u ) > 0 be non-decreasing and σu (k) =
ck c
d|k d
u
. Then,
2 (k, )2s 1 ≤ c ν ψ1 (ν )στ −2s (ν ) . s s k ψ1 (u )στ (u ) ν∈K
u∈ F ( K )
k,∈ K
In particular,
k,∈ K
ck c
(k, )2s ≤ M ( K ) |ck |2 στ −2s (k) k s s
with
k∈ K
Remark 1. Let s > 1/2, 0 < ε ≤ 2s − 1 and take
M(K ) =
k∈ F ( K )
1
στ (k)
.
τ = 1 + ε . Then
2 1 + ε ζ (2s)−1 ck f ks 2 ≤ |ck |2 σ1+ε−2s (k) . k∈ K
ε
k∈ K
We use Theorem 1.3 to prove (with no analysis on the polydisc as in [1]) the following almost everywhere convergence results for functions with bounded variation.
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
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M.J.G. Weber / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
3
Theorem 1.4. Let f ∈ BV(T), f , 1 = 0. Assume that
ck2
k ≥3
(log log k)4 < ∞. (log log log k)2
Then the series
k ck f k
(2)
converges almost everywhere.
Remark 2. This slightly improves Theorem 3 in [1] (nk = k), where it was assumed that the series converges for some γ > 4.
∞
2 γ k=1 ck (log log k)
We will also prove the following rather delicate result where multipliers have arithmetical factors. Theorem 1.5. Let f ∈ BV(T), f , 1 = 0. Assume that for some real b > 0,
k ≥3
ck2 (log log k)2+b σ−1+
Then the series
k ck f k
1
(log log k)b/3
(k) < ∞.
(3)
converges almost everywhere.
These results and some others, notably on the Riemann Zeta function, are proved in [8]. In particular, the following -result is established. Theorem 1.6. Let σ > 1/2. There exist a positive constant c σ depending on any integer ν ≥ 2 such that max[k,]|ν ((kk∨) ,) ≥ c σ , and 0 ≤ ε < σ , we have
max |ζ (σ + it )| ≥ c ζ (2σ )
σ−s+ε (n)2 1/2
1
σ−2ε (ν )
1≤t ≤ T
n 2ε
n|ν
σ only and a positive absolute constant c, such that for
,
whenever ν and T are such that
σ−ε (ν )σ1−σ −ε (ν ) log(ν T )
σ−s+ε (n)2
n|ν
≤
ζ (2σ )1/2 4
n2ε
T ( 2 σ −1 ) .
By taking ν a product of primes, it is easy to recover Theorem 3.3 in [5]. We only sketch the proof of Theorem 1.4, which uses Theorem 1.3. 2. Proof of Theorem 1.4 jB
Choose N j = ee , with B = 2β/δ and δ is a (small) positive real. Let β > 1. Write
ck f k =
N j ≤k< N j +1
J
ck R k +
N j ≤k< N j +1
J
c k rk ,
N j ≤k< N j +1
J
where R J (x) = =1 sin 2π x , r J (x) = f (x) − R J (x) ( J being defined later as a function of j). As f ∈ BV(T), a j = O ( j −1 ), and so by Carleson–Hunt’s maximal inequality,
sup
N j ≤ u ≤ v ≤ N j +1
ck R k 2 ≤ C (log J ) u ≤k≤ v
ck2
1/2
.
N j ≤ u ≤ N j +1
We now combine our Theorem 1.3 with the (ε , 1 − ε ) argument introduced in [1]. Let 0 < ε < 1/2. From the bound J
δk, :=
1 i, j> J jk=i
and since
ij
≤ C min
J 2 u ≤k≤ v ck rk 2
=
(k, ) (k, )2 (k, ) ε (k, )2 1−ε C 1−ε /2 1−ε /2 ≤C , ≤ ε fk , f , (k ∨ ) J k (k ∨ ) J k J
J u ≤k,≤ v ck c δk, ,
we get, choosing
τ = 1 + ε , next using Gronwall’s estimate,
2
(log N j +1 )2ε C C 1−ε /2 J 2 2 ≤ C , c k r k 2 ≤ ε |ck | f k c σ ( k ) ≤ exp − 1 + 2 ε k 2 J ε Jε ε Jε 2ε log log N j +1 u ≤k≤ v
u ≤k≤ v
u ≤k≤ v
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
where
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M.J.G. Weber / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
4
is some positive number. By a well-known variant of Rademacher–Menshov’s maximal inequality,
sup
N j ≤ u ≤ v ≤ N j +1
(log N j +1 )2ε C J 2 2 ck rk 2 ≤ ( log N ) exp j + 1 ε
ε J ε = (log N j+1 )2 exp
Choose
sup
N j ≤ u ≤ v ≤ N j +1
εJ
u ≤k≤ v
2ε log log N j +1
(log N j+1 )2ε ε log log N j+1
2 ck f k 2 ≤ C u ≤k≤ v
with
ε=
ck2
N j ≤ u ≤ N j +1
log log log N j +1 . 2 log log N j +1
ck2 .
N j ≤k≤ N j +1
)2
(log log N
Then log J ≤ C (log log log jN+1 ) , and by combining j +1
(log log k)4 . (log log log k)2
(4)
N
The assumption made implies that the oscillation of the sequence { k=1 ck f k , N ≥ 1} around the subsequence { j ≥ 1} tends to zero almost everywhere. Now, by Tchebycheff’s inequality,
sup
λ
N j ≤ u ≤ v ≤ N j +1
J ck rk > j −β ≤ C j 2β u ≤k≤ v
ck2 ≤ C
N j ≤k≤ N j +1
N j
c f , k=1 k k
ck2 (log log k).
N j ≤ u ≤ N j +1
J Borel–Cantelli’s lemma implies that the series j| N j
J 2
ck R k 2 ≤ ζ (2)
N j
|ck ||c |
H (k, )2 ≤ 4ζ (2) log J k
ck2 σ−1 (k)
μ=h J μ−1 ≤k≤ J μ+2
N j
(k∨)≤ J (k∧)
≤C
ck2
J −1 N j
(log log k)2 log log log k
σ−1 (k).
(5)
By Tchebycheff’s inequality,
λ
J ck R k > j −β ≤ C j 2β
N j
ck2
J −1 N j
≤C J −1 N
j
ck2 J2
(log log k)2 log log log k
(log log k)2+δ log log log k
σ−1 (k)
σ−1 (k).
Treating separately sums with odd indices and sums with even indices allows us to show, by Borel–Cantelli’s lemma, that the series
j
N j
J
ck R k
converges almost everywhere. This allows us to conclude. Final note. In a very recent work, Lewko and Radziwill (arXiv:1408.2334v1) proposed a new approach to Gál’s theorem. They could also reduce the condition γ > 4 in Remark 2 to γ > 2. This naturally includes our Theorem 1.4, but not our Theorem 1.5 with arithmetical multipliers. Further, the new argument we introduced in the proof of Theorem 1.4 suggests a possibility to improve Lewko and Radziwill’s convergence condition by requiring only that k ck2 (log log k)2 /(log log log k)2 < ∞. References [1] C. Aistleitner, I. Berkes, K. Seip, GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc. 17 (6) (2015) 1517–1546. [2] C. Aistleitner, I. Berkes, K. Seip, M. Weber, Convergence of series of dilated functions and spectral norms of GCD matrices, Acta Arith. 168 (3) (2015) 221–246. [3] I. Berkes, M. Weber, On the convergence of ck f (nk x), Mem. Amer. Math. Soc. 201 (943) (2009), vi+72. [4] I. Berkes, M. Weber, On series of dilated functions, Q. J. Math. 65 (1) (2014) 2552. [5] T. Hilberdink, An arithmetical mapping and applications to -results for the Riemann Zeta function, Acta Arith. 139 (2009) 341–367. [6] P. Lindqvist, K. Seip, Note on some greatest common divisor matrices, Acta Arith. LXXXIV (2) (1998) 149–154. [7] M. Weber, On systems of dilated functions, C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1261–1263. [8] M. Weber, An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function, http://arxiv.org/abs/1407.2722v2, 2014.
[m3G; v1.159; Prn:26/08/2015; 8:57] P.1 (1-4)
C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
Contents lists available at ScienceDirect
C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com
Number theory/Harmonic analysis
On convergence almost everywhere of series of dilated functions Sur la convergence presque partout des séries de fonctions dilatées Michel J.G. Weber IRMA, 10, rue du Général-Zimmer, 67084 Strasbourg cedex, France
a r t i c l e
i n f o
Article history: Received 28 July 2014 Accepted after revision 27 July 2015 Available online xxxx Presented by Jean-Pierre Kahane
a b s t r a c t
2 2iπ x Let f (x) = , where ∈Z a e k≥1 ak d(k) < ∞ and d(k) = d|k 1 and let f n (x) = f (nx). We show by using a new decomposition of squared sums that, for any K ⊂ N ∞ ∞ sin 2π jx 2 finite, k∈ K ck f k 22 ≤ ( m=1 am d(m)) k∈ K ck2 d(k2 ). If f s (x) = , s > 1/2, by j =1 js only using elementary Dirichlet convolution calculus, we show that for 0 < ε ≤ 2s − 1, ζ (2s)−1 k∈ K ck f ks 22 ≤ 1+ε ε ( k∈ K |ck |2 σ1+ε−2s (k)), where σh (n) = d|n dh . From this, we
deduce that if f ∈ BV(T), f , 1 = 0 and
∞
4 2 (log log k) k=1 ck (log log log k)2
< ∞, then the series
k ck f k
converges almost everywhere. This slightly improves a recent result, depending on a fine ∞ analysis on the polydisc ([1], th. 3) (nk = k), where it was assumed that k=1 ck2 (log log k)γ converges for some γ > 4. © 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
2 Soit f (x) = ∈Z a e2iπ x telle que la série k≥1 ak d(k) où d(k) = d|k 1 converge, et soit f n ( x) = f ( nx ) . Nous montrons à l’aide d’une nouvelle décomposition des sommes carrées ∞ 2 que k∈ K ck f k 22 ≤ ( m=1 am d(m)) k∈ K ck2 d(k2 ), pour tout ensemble fini d’entiers K .
Si f s (x) =
∞
j =1
sin 2π jx , js
s > 1/2, nous montrons aussi, par un calcul simple sur les
ε ≤ 2s − 1 et σh (n) = f , 1 = 0, si la série
∞
ε 2 ck f ks 22 ≤ 1+ ε ( k∈ K |ck | σ1+ε−2s (k)), où 0 < h d . Nous en déduisons que, pour tout f ∈ BV(T) telle que d|n
convolutions de Dirichlet, que ζ (2s)−1
4 2 (log log k) k=1 ck (log log log k)2
k∈ K
converge, alors la série
k ck f k
converge presque
partout. Cela améliore un résultat récent, dépendant analyse fine sur le polydisque ∞ d’une 2 γ ([1], th. 3) (nk = k), où l’on suppose que la série k=1 ck (log log k) converge pour un réel γ > 4. © 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.crma.2015.07.010 1631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
[m3G; v1.159; Prn:26/08/2015; 8:57] P.2 (1-4)
M.J.G. Weber / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
2
1. Introduction – main result One of the oldest and most central problems ∞ in the theory of systems of dilated sums is the study of the convergence in norm or almost everywhere of the series k=1 ck f (nk x), where f is a periodic function f and N = {nk , k ≥ 1} a sequence of positive integers (see [3]). Our main concern is the search of individual conditions ensuring convergence, a barely investigated part of the theory. We use an arithmetical approach based on elementary Dirichlet convolution calculus and on a new decomposition of squared sums, continuing the work in [7,4]. We show that this approach is strong enough to recover and even slightly improve a recent a.e. convergence result [1] (Theorem 3) in the case N = N without using analysis on the polydisc. Results in [1] were recently developed in [2]. Our approach is also in the spirit of the work of Hilberdink [5] on some arithmetical mappings and extrema linked to arithmetical functions, with applications ∞ to -results of the Riemann Zeta function. Denote e (x) = e2iπ x , en (x) = e (nx), n ≥ 1. Let T = R/Z = [0, 1[. Let f (x) ∼ j =1 a j e j (x). Let f n (x) = f (nx), n ∈ N. We assume throughout that
f ∈ L 2 (T),
f , 1 = 0.
(1)
A key preliminary step naturally consists in searching bounds of k∈ K ck f k 2 integrating in their formulation the arithmetical structure of K . That question has received a satisfactory answer only for specific cases. We state our mean results. Let d(n) be the divisor function, namely the number of divisors of n. Throughout, K denotes a finite set of natural numbers. Theorem 1.1. Assume that
∞
2 m=1 am d(m)
< ∞. Then,
∞ 2 2 2 ck f k ≤ a2 d(m) c d(k ). 2
k∈ K
m
k
m =1
k∈ K
In [7], using Hooley’s Delta function, we recently showed a similar estimate however restricted to sets K such that r r +1 K ⊂]e , e ] for some integer r. Theorem 1.1 is deduced from a more general result. Introduce the necessary notation. Let ∞ A k = ν =1 a2ν k . Let ζh be defined by ζh (n) = nh for all positive n. Let θ(n) denotes the number of squarefree divisors of n. Given K ⊂ N, we note F ( K ) = {d ≥ 1; ∃k ∈ K : d|k}. If K is factor closed (d|k ⇒ d ∈ K for all k ∈ K ), then F ( K ) = K . Theorem 1.2. Let ψ be any arithmetical function taking only positive values. Then,
2 ck f k 2 ≤ B ck2 ψ ∗ ζ0 (k), k∈ K
where
B = sup
A dk
d∈ F ( K )
k∈ K
k∈ K d|k
ψ( dk )
k θ( ) < ∞. d
Here ∗ denotes the Dirichlet convolution. By choosing ψ = θ and since ψ ∗ ζ0 (k) = d(k2 ), we check that B ≤ ∞ sin 2π jx 2 s , s > 1/2 m=1 am d(m), whence Theorem 1.1. Consider now the class of functions introduced in [6], f (x) = j =1 js
∞
2s
and recall that f ks , f s = ζ (2s) (kk,) s s
where (k, ) = gcd(k, ).
Theorem 1.3. Let s > 0, 0 ≤ τ ≤ 2s. Let also ψ1 (u ) > 0 be non-decreasing and σu (k) =
ck c
d|k d
u
. Then,
2 (k, )2s 1 ≤ c ν ψ1 (ν )στ −2s (ν ) . s s k ψ1 (u )στ (u ) ν∈K
u∈ F ( K )
k,∈ K
In particular,
k,∈ K
ck c
(k, )2s ≤ M ( K ) |ck |2 στ −2s (k) k s s
with
k∈ K
Remark 1. Let s > 1/2, 0 < ε ≤ 2s − 1 and take
M(K ) =
k∈ F ( K )
1
στ (k)
.
τ = 1 + ε . Then
2 1 + ε ζ (2s)−1 ck f ks 2 ≤ |ck |2 σ1+ε−2s (k) . k∈ K
ε
k∈ K
We use Theorem 1.3 to prove (with no analysis on the polydisc as in [1]) the following almost everywhere convergence results for functions with bounded variation.
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
[m3G; v1.159; Prn:26/08/2015; 8:57] P.3 (1-4)
M.J.G. Weber / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
3
Theorem 1.4. Let f ∈ BV(T), f , 1 = 0. Assume that
ck2
k ≥3
(log log k)4 < ∞. (log log log k)2
Then the series
k ck f k
(2)
converges almost everywhere.
Remark 2. This slightly improves Theorem 3 in [1] (nk = k), where it was assumed that the series converges for some γ > 4.
∞
2 γ k=1 ck (log log k)
We will also prove the following rather delicate result where multipliers have arithmetical factors. Theorem 1.5. Let f ∈ BV(T), f , 1 = 0. Assume that for some real b > 0,
k ≥3
ck2 (log log k)2+b σ−1+
Then the series
k ck f k
1
(log log k)b/3
(k) < ∞.
(3)
converges almost everywhere.
These results and some others, notably on the Riemann Zeta function, are proved in [8]. In particular, the following -result is established. Theorem 1.6. Let σ > 1/2. There exist a positive constant c σ depending on any integer ν ≥ 2 such that max[k,]|ν ((kk∨) ,) ≥ c σ , and 0 ≤ ε < σ , we have
max |ζ (σ + it )| ≥ c ζ (2σ )
σ−s+ε (n)2 1/2
1
σ−2ε (ν )
1≤t ≤ T
n 2ε
n|ν
σ only and a positive absolute constant c, such that for
,
whenever ν and T are such that
σ−ε (ν )σ1−σ −ε (ν ) log(ν T )
σ−s+ε (n)2
n|ν
≤
ζ (2σ )1/2 4
n2ε
T ( 2 σ −1 ) .
By taking ν a product of primes, it is easy to recover Theorem 3.3 in [5]. We only sketch the proof of Theorem 1.4, which uses Theorem 1.3. 2. Proof of Theorem 1.4 jB
Choose N j = ee , with B = 2β/δ and δ is a (small) positive real. Let β > 1. Write
ck f k =
N j ≤k< N j +1
J
ck R k +
N j ≤k< N j +1
J
c k rk ,
N j ≤k< N j +1
J
where R J (x) = =1 sin 2π x , r J (x) = f (x) − R J (x) ( J being defined later as a function of j). As f ∈ BV(T), a j = O ( j −1 ), and so by Carleson–Hunt’s maximal inequality,
sup
N j ≤ u ≤ v ≤ N j +1
ck R k 2 ≤ C (log J ) u ≤k≤ v
ck2
1/2
.
N j ≤ u ≤ N j +1
We now combine our Theorem 1.3 with the (ε , 1 − ε ) argument introduced in [1]. Let 0 < ε < 1/2. From the bound J
δk, :=
1 i, j> J jk=i
and since
ij
≤ C min
J 2 u ≤k≤ v ck rk 2
=
(k, ) (k, )2 (k, ) ε (k, )2 1−ε C 1−ε /2 1−ε /2 ≤C , ≤ ε fk , f , (k ∨ ) J k (k ∨ ) J k J
J u ≤k,≤ v ck c δk, ,
we get, choosing
τ = 1 + ε , next using Gronwall’s estimate,
2
(log N j +1 )2ε C C 1−ε /2 J 2 2 ≤ C , c k r k 2 ≤ ε |ck | f k c σ ( k ) ≤ exp − 1 + 2 ε k 2 J ε Jε ε Jε 2ε log log N j +1 u ≤k≤ v
u ≤k≤ v
u ≤k≤ v
JID:CRASS1 AID:5560 /FLA Doctopic: Number theory
where
[m3G; v1.159; Prn:26/08/2015; 8:57] P.4 (1-4)
M.J.G. Weber / C. R. Acad. Sci. Paris, Ser. I ••• (••••) •••–•••
4
is some positive number. By a well-known variant of Rademacher–Menshov’s maximal inequality,
sup
N j ≤ u ≤ v ≤ N j +1
(log N j +1 )2ε C J 2 2 ck rk 2 ≤ ( log N ) exp j + 1 ε
ε J ε = (log N j+1 )2 exp
Choose
sup
N j ≤ u ≤ v ≤ N j +1
εJ
u ≤k≤ v
2ε log log N j +1
(log N j+1 )2ε ε log log N j+1
2 ck f k 2 ≤ C u ≤k≤ v
with
ε=
ck2
N j ≤ u ≤ N j +1
log log log N j +1 . 2 log log N j +1
ck2 .
N j ≤k≤ N j +1
)2
(log log N
Then log J ≤ C (log log log jN+1 ) , and by combining j +1
(log log k)4 . (log log log k)2
(4)
N
The assumption made implies that the oscillation of the sequence { k=1 ck f k , N ≥ 1} around the subsequence { j ≥ 1} tends to zero almost everywhere. Now, by Tchebycheff’s inequality,
sup
λ
N j ≤ u ≤ v ≤ N j +1
J ck rk > j −β ≤ C j 2β u ≤k≤ v
ck2 ≤ C
N j ≤k≤ N j +1
N j
c f , k=1 k k
ck2 (log log k).
N j ≤ u ≤ N j +1
J Borel–Cantelli’s lemma implies that the series j| N j
J 2
ck R k 2 ≤ ζ (2)
N j
|ck ||c |
H (k, )2 ≤ 4ζ (2) log J k
ck2 σ−1 (k)
μ=h J μ−1 ≤k≤ J μ+2
N j
(k∨)≤ J (k∧)
≤C
ck2
J −1 N j
(log log k)2 log log log k
σ−1 (k).
(5)
By Tchebycheff’s inequality,
λ
J ck R k > j −β ≤ C j 2β
N j
ck2
J −1 N j
≤C J −1 N
j
ck2 J2
(log log k)2 log log log k
(log log k)2+δ log log log k
σ−1 (k)
σ−1 (k).
Treating separately sums with odd indices and sums with even indices allows us to show, by Borel–Cantelli’s lemma, that the series
j
N j
J
ck R k
converges almost everywhere. This allows us to conclude. Final note. In a very recent work, Lewko and Radziwill (arXiv:1408.2334v1) proposed a new approach to Gál’s theorem. They could also reduce the condition γ > 4 in Remark 2 to γ > 2. This naturally includes our Theorem 1.4, but not our Theorem 1.5 with arithmetical multipliers. Further, the new argument we introduced in the proof of Theorem 1.4 suggests a possibility to improve Lewko and Radziwill’s convergence condition by requiring only that k ck2 (log log k)2 /(log log log k)2 < ∞. References [1] C. Aistleitner, I. Berkes, K. Seip, GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc. 17 (6) (2015) 1517–1546. [2] C. Aistleitner, I. Berkes, K. Seip, M. Weber, Convergence of series of dilated functions and spectral norms of GCD matrices, Acta Arith. 168 (3) (2015) 221–246. [3] I. Berkes, M. Weber, On the convergence of ck f (nk x), Mem. Amer. Math. Soc. 201 (943) (2009), vi+72. [4] I. Berkes, M. Weber, On series of dilated functions, Q. J. Math. 65 (1) (2014) 2552. [5] T. Hilberdink, An arithmetical mapping and applications to -results for the Riemann Zeta function, Acta Arith. 139 (2009) 341–367. [6] P. Lindqvist, K. Seip, Note on some greatest common divisor matrices, Acta Arith. LXXXIV (2) (1998) 149–154. [7] M. Weber, On systems of dilated functions, C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1261–1263. [8] M. Weber, An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function, http://arxiv.org/abs/1407.2722v2, 2014.