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Dynamical Diophantine approximation of beta expansions of formal Laurent series
Finite Fields and Their Applications 34 (2015) 176–191
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Finite Fields and Their Applications www.elsevier.com/locate/ffa
Dynamical Diophantine approximation of beta expansions of formal Laurent series Chao Ma a , Shuailing Wang b,∗ a
Faculty of Information Technology and Department of General Education, Macau University of Science and Technology, Macau b Department of Mathematics, South China University of Technology, Guangzhou, 510640, PR China
a r t i c l e
i n f o
Article history: Received 3 January 2015 Received in revised form 30 January 2015 Accepted 31 January 2015 Available online xxxx Communicated by Arne Winterhof MSC: 11K55 28A80 Keywords: Dynamical Diophantine approximation β-Expansion Formal Laurent series Hausdorff dimension
a b s t r a c t Let Fq be a finite field with q elements and Fq ((X −1 )) be the field of the formal Laurent series with an indeterminant X. Let Tβ be the β-transformation defined on Fq ((X −1 )). This paper is concerned with the size of the following dynamically defined limsup set n−1 x ∈ Fq ((X −1 )) : Tβn x − y0 < q −(f (x)+···+f (Tβ x)) , for infinitely many n ∈ N , for any given y0 ∈ Fq ((X −1 )), where f is a positive continuous function and · is the norm, defined on Fq ((X −1 )). We show that its Hausdorff dimension is given as the solution s0 to the pressure function equation P(−s(deg β + f ), Tβ ) = 0. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (C. Ma), [email protected] (S. Wang). http://dx.doi.org/10.1016/j.ffa.2015.01.011 1071-5797/© 2015 Elsevier Inc. All rights reserved.
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1. Introduction Let (X, d) be a metric space and T : X → X be a transformation. Suppose that X is equipped with a T -invariant Borel probability measure μ. If T is further assumed to be ergodic with respect to the measure μ, then for any ball B in X of positive measure, the set x ∈ X : T n x ∈ B, for infinitely many n ∈ N
has full μ-measure. Chernov and Kleinbock [2] investigated the case when the fixed set B is replaced by a sequence of measurable sets {Bn }n≥1 in a general setting. More precisely, they concerned the size from the sense of the measure of the set (1.1) x ∈ X : T n x ∈ Bn , for infinitely many n ∈ N . This is called the dynamical Borel–Cantelli lemma [2] (see also [8,17]). Because of its analogy to the classic Diophantine approximation, this dynamically defined limsup set is also termed as the dynamical Diophantine approximation [4] (see also [5,26]). The set (1.1) is also called the shrinking target problem [11] and many mathematicians studied the size in the sense of the dimension of the set (1.1), see [1,15,18,24]. In this note, we consider the size of the set defined in (1.1) for β-expansions in the field of formal Laurent series, when Bn is a sequence of balls with given center. Let us first recall the field of formal Laurent series Fq ((X −1 )) and the β-expansion defined on it. Let Fq be a finite field with q elements, and Fq [X] be the ring of polynomials with coefficients in Fq and Fq (X) be the field of fractions. Then Fq ((X −1 )) =
+∞
xn X −n : xn ∈ Fq and n0 ∈ Z .
n=n0
The norm of x is defined to be x = q deg(x) , where deg(x) = − inf{n ∈ Z : xn = 0}, called the degree of x and deg(0) = −∞ with the convention. It is well known that · is a non-Archimedean norm on the field Fq ((X −1 )), that is, (1) (2) (3) (4)
x ≥ 0 with x = 0 if and only if x = 0; xy = x · y; αx + βy ≤ max(x, y) (∀α, β ∈ F); For all α, β ∈ F, α = 0, β = 0, if x = y, then αx + βy = max(x, y).
This norm induces a metric ρ as ρ(x, y) = x − y satisfying: (i) every point of a ball may be considered as the center of the ball; and (ii) if two balls intersect, the one with
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larger radius must contain the other. Moreover, Fq ((X −1 )) is a complete metric space under the metric ρ. Denote by I = x ∈ F((X −1 )) : x < 1 the unit ball of Fq ((X −1 )). The set I is isomorphic to n≥1 Fq and is an abelian compact group. As a consequence, there exists a unique normalized Haar measure μ on I given by
μ B(a, q −r ) = q −r , where B(a, q −r ) = x ∈ Fq ((X −1 )) : x − a < q −r is the disc of center a ∈ Fq ((X −1 )) and radius q −r with r ∈ Z. Note that μ(I) = 1 and (I, B (I), μ) is a probability space, where B (I) is a Borel field on I. Every x ∈ Fq ((X −1 )) has a unique (Artin) decomposition as x = [x] + {x}, where the integral part [x] of x belongs to Fq [X] and the fractional part {x} of x belongs to I. Let β ∈ Fq ((X −1 )) with β > 1. The β-transformation Tβ on I is defined as Tβ x = βx − [βx]. Then every x ∈ I can be represented by x=
ε1 (x) ε2 (x) εn (x) + + ··· + + ··· β β2 βn
(1.2)
with ε1 (x) = [βx] and εn (x) = ε1 (Tβn−1 (x)) for all n ≥ 2, called β-digits of x. The formula (1.2) is called the β-expansion of x, which was introduced by Scheicher [22], Hbaib and Mkaouar [10] independently. Theorem 1.1. (See [22].) An infinite sequence (εn )n≥1 in Fq [X] is the β-expansion of x ∈ I if and only if εn < β for all n ≥ 1. Denote by Σβ = {ε ∈ Fq [X] : ε < β} the set of possible β-digits of all formal Laurent series and by Σnβ := {ε ∈ Fq [X] : ε < β}n the set of possible words of length n. It is known that #Σβ = β, that is, the number of possible β-digits is β. Li, Wu and Xu [14] proved that Tβ is invariant and ergodic with respect to the Haar measure μ, and showed some limit theorems for β-digits and the Hausdorff dimensions of some exceptional sets according to the limit theorem. Wang [28] considered the metric theorem and exceptional sets about the approximation order of the β-expansion, that is, the convergent speed of the series in formula (1.2). Jellali, Mkaouar and Scheicher [12] (see also [9,23]) characterized the series whose β-expansion is purely periodic when β is a Pisot unit. Fan, Wang and Zhang [6] calculated the Hausdorff dimensions of the quantitative recurrence and hitting sets of β-transformation Tβ on I. Recently Lü [16] studied the Diophantine approximation problem for Tβ . The β-expansion on I was introduced as the analogue of that on real line. The β-expansion of the real numbers was given by Rényi [21] in 1957 and was studied extensively since then (see [13,19,20,25] etc).
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The metric theory about the Haar measure μ of the set defined in (1.1) falls into the general setting of Fernández, Melián and Pestana [7] which indicates that μ
x ∈ I : Tβn x ∈ Bn , i.o. n ∈ N =1
if and only if
μ(Bn ) = ∞.
n≥1
Here i.o. is a brevity for infinitely often. While on the other hand, it is clear but not trivial that there exist points violating this law. In this case, Hausdorff dimension serves as a tool to classify the size of zero-measure sets. This note is devoted to determining the Hausdorff dimensions of the following zero-measure sets. Let f be a positive continuous function defined on I and y0 be some fixed point in I. Define the limsup set S(f ) as S(f ) := x ∈ I : Tβn x − y0 < q −(Sn f )(x) , i.o. n ∈ N where Sn f denotes the ergodic sum, i.e., (Sn f )(x) =
n−1 i=0
f (Tβi x). We show that:
Theorem 1.2. The Hausdorff dimension of the set S(f ) is given as the solution s0 to the pressure function equation P(−s(deg β + f ), Tβ ) = 0.
(1.3)
For the definition and properties of pressure function, see Chapter 9 in Walters [27]. Remark. From Proposition 2.4 below, we know that the pressure equation (1.3) has the unique solution. 2. Preliminary Definition 2.1. For any given block (ε1 , ε2 , . . . , εn ) ∈ Σnβ , In (ε1 , ε2 , . . . , εn ) = {x ∈ I : ε1 (x) = ε1 , ε2 (x) = ε2 , . . . , εn (x) = εn } is called an n-th cylinder of the β-expansion. We denote by In (x) the n-th cylinder containing x. The measures μ(In (ε1 , ε2 , . . . , εn )) are very regular for any (ε1 , ε2 , . . . , εn ) ∈ Σnβ from the following proposition.
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Proposition 2.2. (See [14].) For any cylinder In (ε1 , ε2 , · · · , εn ), we have In (ε1 , ε2 , · · · , εn ) = B
ε2 εn 1 ε1 + 2 + ··· + n, β β β βn
.
(2.1)
As a consequence, μ(In (ε1 , ε2 , · · · , εn )) = β−n . Proposition 2.3. The β-transformation Tβ is continuous on I. Proof. For any x, y ∈ I, note that Tβ y −Tβ x = β ·y −x, then the result holds. 2 The topological pressure of a continuous map can be defined in different equivalent ways such as open covering, spanning set and separated set (see [27]). Since {I1 (ε) : ε ∈ Σβ } is a generator for Tβ , for any potential function ϕ ∈ C(Fq ((X −1 )), R), the topological pressure can be defined as P(ϕ, Tβ ) = lim
n→∞
1 logq n
q (Sn ϕ)(x) ,
(2.2)
(ε1 ,...,εn )∈Σn β
where x ∈ In (ε1 , . . . , εn ) and the limit does not depend on the choice of x. From (2.2) and the positiveness of f , we have the following. Proposition 2.4. The pressure function P(−s(deg β + f ), Tβ ) is strictly decreasing on s. 3. Proof of Theorem 1.2 We will prove Theorem 1.2 in this section. Naturally, the proof is divided into two parts by the estimations from above and below. Given s > 0 and a subset E of I, the s-Hausdorff measure is defined as Hs (E) = lim
⎧ ⎨ inf
δ→0 ⎩
j
|Bj |s
⎫ ⎬ ⎭
,
where the infimum is taken over all covers of E by balls Bj with diameter at most δ, and | · | denotes the diameter of a set. The Hausdorff dimension of the set E is given by dimH (E) = inf{s ≥ 0 : Hs (E) = 0}. 3.1. Upper bound The upper bound of dimH S(f ) can be obtained by considering the natural covering system. Evidently,
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S(f ) =
∞ ∞
J(ε1 , · · · , εn ),
181
(3.1)
N =1 n=N (ε1 ,···,εn )∈Σn β
where J(ε1 , · · · , εn ) = x ∈ In (ε1 , · · · , εn ) : Tβn x − y0 < q −(Sn f )(x) . Then, for any N ≥ 1, {J(ε1 , · · · , εn ) : (ε1 , · · · , εn ) ∈ Σnβ , n ≥ N } is a natural cover of the set S(f ). Note that sup {(Sn f )(x) − (Sn f )(y) : In (x) = In (y)} ≤
n
sup {|f (x) − f (y)| : Ii (x) = Ii (y)} ,
i=1
since the continuity of f implies n
sup {|f (x) − f (y)| : Ii (x) = Ii (y)} = o(n);
i=1
for any δ > 0, there exists an integer N ∈ N such that for all n ≥ N , sup (Sn f )(x) − (Sn f )(y) : In (x) = In (y) ≤ nδ. Therefore, if we replace x by zn :=
εn (x) ε1 (x) + ··· + ∈ In (x) β βn
in (Sn f )(x), there is a loss at most nδ, that is, |(Sn f )(zn ) − (Sn f )(x)| ≤ nδ.
(3.2)
J(ε1 , · · · , εn ) ⊂ x ∈ In (ε1 , · · · , εn ) : Tβn x − y0 < q −(Sn f )(zn )+nδ .
(3.3)
Thus, we have
While, it is clear that the set in the right side of (3.3) is the ball B
εn y0 −n deg β −(Sn f )(zn )+nδ ε1 . + ··· + n + n,q ·q β β β
Hence we obtain, for large n, that J(ε1 , · · · , εn ) ≤ q −n deg β−(Sn f )(zn )+nδ .
(3.4)
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Combining (3.1) and (3.4), for any s > s0 , we have Hs (S(f )) ≤ lim inf N →∞
≤ lim inf N →∞
∞
J(ε1 , · · · , εn )s
n=N (ε1 ,···,εn )∈Σn β ∞
q (−n deg β−(Sn f )(zn )+nδ)s .
n=N (ε1 ,···,εn )∈Σn β
By Proposition 2.4, −αs := P(−s(deg β + f ), Tβ ) < 0 since s > s0 . From (2.2), we know that when n is large enough,
q (−n deg β−(Sn f )(zn ))s ≤ q −
αs 2
n
.
(ε1 ,···,εn )∈Σn β
Therefore, H (S(f )) ≤ lim inf s
N →∞
∞
q −(
αs 2
−δs)n
,
n=N
which implies Hs (S(f )) < +∞ by choosing δ <
αs 2s .
Thus dimH S(f ) ≤ s0 .
3.2. Lower bound For any (ε1 , · · · , εn ) ∈ Σnβ , in need of picking a point zn in In (ε1 , · · · , εn ), we always take, with no extra announcement, zn =
εn ε1 + ··· + n. β β
Lemma 3.1. Let g be a positive continuous function and y0 ∈ I. Then the Hausdorff dimension s of the following set S(g) := x ∈ I : Tβn x − y0 < q −(Sn g)(zn ) with zn ∈ In (x), i.o. n ∈ N fulfills the pressure equation P(−s(deg β + g), Tβ ) = 0. We claim that Lemma 3.1 implies Theorem 1.2. Indeed, take g = f + (deg β + f )δ for any δ > 0. By the continuity of f , it is clear that S(g) ⊂ S(f ) since (Sn g)(zn ) ≥ (Sn f )(x) from (3.2). Let s0 be the solution of Eq. (1.3), then s0 (deg β + g), Tβ 0 = P(−s0 (deg β + f ), Tβ ) = P − 1+δ
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by noting f =
g−δ deg β . 1+δ
183
As a consequence, dimH S(f ) ≥ dimH S(g) =
s0 , 1+δ
which gives a good lower bound of the Hausdorff dimension of the set S(f ) by letting δ → 0. Therefore, the left is to prove Lemma 3.1. The following mass distribution principle will be applied to obtain the lower bound of dimH S(g). Proposition 3.2. (See Falconer [3].) Let E be a Borel measurable set in a metric space X and μ be a Borel measure with μ(E) > 0. Assume that there exist two positive constants c, η such that, for any set U with diameter diam(U ) less than η, μ(U ) ≤ c diam(U )s , then dimH E ≥ s. Directly by this proposition, we are led to the following three steps: firstly, construct a Cantor subset C∞ of S(g); secondly, define a measure μ supported on C∞ , and at last, estimate the Hölder exponent of the measure μ. 3.2.1. Construction of a Cantor set First we outline the idea for the construction of a Cantor subset of S(g). Main strategy: By the algorithm of β-expansion, we know that every x ∈ I has a unique symbolic representation by the sequence of its digits (ε1 (x), ε2 (x), · · ·). Such a representation makes it much convenient to achieve that two points x, y are close enough just by letting x, y have the same prefix for a long time in their symbolic representations. More precisely, to make sure that x ∈ B(y, r) for some r > 0, we choose the integer t such that |It (y)| ≤ r < |It−1 (y)| and x can be chosen as any element in It (y). For a digit block (ε1 , · · · , εn ) ∈ Σnβ , define tn = tn (ε1 , · · · , εn ) to be the integer such that |Itn (y0 )| ≤ q −(Sn g)(zn ) < |Itn −1 (y0 )|,
(3.5)
n ε where zn = j=1 βjj . Now we construct a Cantor subset of S(g). In the sequel, we denote by (b1 , b2 , · · · , ) the sequence of the digits of y0 in its β-expansion. Choose a largely sparse sequence {mk }k≥1 with m1 1,
k = o(1), m1 + · · · + mk−1 mk (k ≥ 1). mk
(3.6)
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(1)
(1)
1 Level 1 of the Cantor set. Take n1 = m1 . For each (ε1 , · · · , εm1 ) ∈ Σm β , for simplicity of notation, we denote by t1 the integer defined in (3.5), i.e. t1 is the integer satisfying
q −t1 deg β ≤ q −(Sn1 g)(z1 ) < q −(t1 −1) deg β , (1)
where z1 =
ε1 β
+ ··· +
ε(1) n1 β n1
(1)
(1)
∈ In1 (ε1 , · · · , εn1 ). (1)
(1)
Note that the integer t1 depends on the sequence (ε1 , · · · , εm1 ), but we omit this dependence in the notation for simplicity. Then the first level of the Cantor set is defined as
(1) (1) m1 (1) C (1) = In1 +t1 ε1 , · · · , ε(1) , b , · · · , b , · · · , ε ) ∈ Σ : (ε . t1 m1 1 m1 1 β Other levels of the Cantor set are defined by an induction. Level k of the Cantor set. Assume that the (k − 1)-th level of the Cantor set has been well constructed. Now fix a cylinder Jk−1 = Ink−1 +tk−1 (E (k−1) ) ∈ C (k − 1) and write (k) (k) k nk = nk−1 + tk−1 + mk . For any (ε1 , · · · , εmk ) ∈ Σm β , determine tk by (3.5) for the digit block (k)
(E (k−1) , ε1 , · · · , ε(k) mk ), i.e. tk is the integer satisfying q −tk deg β ≤ q −(Snk g)(zk ) < q −(tk −1) deg β , (k)
(3.7)
(k)
where zk ∈ Ink (E (k−1) , ε1 , · · · , εmk ). The right inequality in (3.7) indicates that tk is bounded from above by nk , more precisely, tk < 1 +
nk g , deg β
(3.8)
where g is the maximal norm of g in C(I, R). Then a subcollection of the second level is given as
(k) (k) mk (k) C (k, Jk−1 ) = Ink +tk E (k−1) , ε1 , · · · , ε(k) . mk , b1 , · · · , btk : (ε1 , · · · , εmk ) ∈ Σβ Then we define the k-th level as C (k) = C (k, Jk−1 ) : Jk−1 ∈ C (k − 1) . The Cantor set. Continuing this process, we obtain a nested sequence {C (k)}k≥1 composed of cylinders, called basic cylinders. And then the desired Cantor set is C∞ =
∞
k=1 Jk ∈C (k)
Jk .
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From the construction of C∞ , we have C∞ ⊂ S(g).
3.2.2. Supporting measure We distribute a probability measure μ on C∞ . For each k ≥ 1, define sk as the solution to the equation
1=
q −s(mk deg β+(Smk g)(zk )) ,
(3.9)
mk
(ε1 ,···,εmk )∈Σβ
k where zk is chosen in Imk (ε1 , · · · , εmk ) for each (ε1 , · · · , εmk ) ∈ Σm β . Comparing the elements zk and zk in (3.7) and (3.9), it should be noticed that
(1)
(k)
(k) zk ∈ Ink (ε1 , · · · , ε(1) m1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk ),
while zk ∈ Imk (ε1 , · · · , ε(k) mk ). (k)
Now we collect some simple observations from these relations which will be used later. One is the following n
Tβ k−1
+tk−1
zk = zk , |(Snk g)(zk ) − (Smk g)(zk )| ≤ mk δ,
(3.10)
where the inequality holds because of (3.6) and (3.8). It is clear that, by the definition of the pressure function, we have Lemma 3.3. lim sk = s0 .
(3.11)
k→∞
For any basic cylinder Jk ∈ C (k), let Jk−1 ∈ C (k − 1) be its mother cylinder, i.e. Jk ∈ C (k, Jk−1 ). Then by the construction of C (k, Jk−1 ), the cylinders Jk and Jk−1 can be expressed as (k)
(k−1) Jk = Ink +tk (E (k−1) , ε1 , · · · , ε(k) ). mk , b1 , . . . , btk ), Jk−1 = Ink−1 +tk−1 (E
The measure of Jk is defined as μ(Jk ) = μ(Jk−1 )
s k
1
q mk deg β+(Smk g)(zk )
where zj ∈ Imj (ε1 , · · · , εmj ), for 1 ≤ j ≤ k. (j)
(j)
=
k
s j
1
j=1
q mj deg β+(Smj g)(zj )
, (3.12)
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Then we extend the definition of the set function μ to be a measure supported on C∞ . For each x ∈ C∞ , by the construction of C∞ , there exist a sequence of digit blocks (k) (k) {ε1 , · · · , εmk }k≥1 and a sequence of integers {tk }k≥1 such that (1)
(k)
(k) x = [ε1 , · · · , ε(1) m1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk , b1 , · · · , btk , · · ·].
Moreover, the integer sequence {tk }k≥1 satisfies (3.7), i.e., q −tk deg β ≤ q −(Snk g)(zk ) < q −(tk −1) deg β , (1)
(1)
(k)
(3.13)
(k)
where zk ∈ Ink (ε1 , · · · , εm1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk ). For each k ≥ 1 and nk−1 + tk−1 < n ≤ nk + tk , we define the measure of In (x) in the following way. (1). When nk−1 + tk−1 < n < nk . Write n = nk−1 + tk−1 + . Then the cylinder In (x) can be expressed as (k)
(k)
In (x) = In (E (k−1) , ε1 , · · · , ε ). Then we define
μ(In (x)) = (k)
(k)
(k)
(k)
(k)
μ(Ink +tk (E (k−1) , ε1 , · · · , ε , ε+1 , · · · , ε(k) mk , b1 , · · · , btk )) mk −
(ε+1 ,···,εmk )∈Σβ
where the summation is taken over all digit blocks (ε+1 , · · · , εmk ) ∈ Σβmk − . Thus, (k)
μ(In (x)) k−1 = j=1
s j
1
q mj deg β+(Smj g)(zj )
(k)
sk
1
m − (k) (k) (ε+1 ,···,εmk )∈Σβ k
q mk deg β+(Smk g)(zk )
. (3.14)
(2). When nk ≤ n ≤ nk + tk ,
μ(In (x)) = μ(Jk ) =
k
s j
1
j=1
q mj deg β+(Smj g)(zj )
.
(3.15)
The cylinders that do not intersect C∞ are allocated zero by the distribution μ. Thus the measure μ is well defined on all cylinders. By Kolmogorov’s extension theorem, the measure μ can be defined on (I, B(I)). It is easy to know that the support of μ is C∞ from the construction of μ.
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3.2.3. Hölder exponent of the measure • Estimation of the μ-measure of cylinders. Of course, we only consider the cylinders with non-zero μ-measure. Given a basic cylinder Jk ∈ C (k), write it as (1)
(k)
(k) Jk = Ink +tk (ε1 , · · · , ε(1) m1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk , b1 , . . . , btk ).
At first, we estimate the length of Jk . It is clear that |Jk | = q −(nk +tk ) deg β = q −(m1 +···+mk ) deg β q −(t1 +···+tk ) deg β . Following from (3.13) on the definition of tk , we have q −k deg β
k
q −(mj deg β+(Snj g)(zj )) ≤ |Jk | ≤
j=1
k
q −(mj deg β+(Snj g)(zj )) .
(3.16)
j=1
Recalling the measure of Jk (see formula (3.12)) and the difference on zj and zj for 1 ≤ j ≤ k (see formula (3.10)), it follows directly that μ(Jk ) ≤ |Jk |s0 −δ .
(3.17)
For any x ∈ C∞ , we will check the Hölder exponent of μ along the cylinders containing x. For each n ≥ n1 + t1 , let k > 1 be the integer such that nk−1 + tk−1 < n ≤ nk + tk . (I). When nk−1 + tk−1 < n < nk , writing n = nk−1 + tk−1 + ,
μ(In (x)) = μ(Jk−1 )
m − (k) (k) (ε+1 ,···,εmk )∈Σβ k
= μ(Jk−1 )
1
s k
1
q mk deg β+(Smk g)(zk )
s k
1
q deg β
≤ μ(Jk−1 )
s k
1
m − (k) (k) (ε+1 ,···,εmk )∈Σβ k
s k
q (mk −) deg β+(Smk g)(zk )
sk
1
q deg β
mk −
(ε+1 ,···,εmk )∈Σβ
q (mk −) deg β+(Smk − g)(Tβ zk )
where the last inequality follows by the simple fact of positiveness of g. Now we estimate the last summation in detail. Recall the definition of sk . 1=
s k
1
m (ε1 ,···,ε ,ε+1 ,···,εmk )∈Σβ k
q mk deg β+(Smk g)(zk )
,
k where zk ∈ Imk (ε1 , · · · , εmk ) for each (ε1 , · · · , εmk ) ∈ Σm β . By Lemma 3.3, there exists 0 ∈ N such that for any , k > 0 , |s − sk | < δ.
,
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When ≤ 0 ,
1≥
q mk deg β+(Smk g)(zk )
mk −
(ε+1 ,···,εmk )∈Σβ
≥
s k
1
1
sk
1
.
q (deg β+g)
q (mk −) deg β+(Smk − g)(Tβ zk )
mk −
(ε+1 ,···,εmk )∈Σβ
Hence we have
sk
1
≤ q (deg β+g)
q (mk −) deg β+(Smk − g)(Tβ zk )
m − (ε+1 ,···,εmk )∈Σβ k
≤ q 0 (deg β+g) := C1 . So, together with the first inequality in (3.17), we get
s k
1
μ(In (x)) ≤ C1 μ(Jk−1 )
s0 −δ
1
≤ C1 q deg β q (nk−1 +tk−1 ) deg β s0 −δ 1 ≤ C1 = C1 |In (x)|s0 −δ . n deg β q
1
s k
q deg β
When > 0 , |s − sk | < δ, so,
1≥
s k
1
mk
(ε1 ,···,ε ,ε+1 ,···,εmk )∈Σβ
=
mk −
(ε+1 ,···,εmk )∈Σβ
×
q mk deg β+(Smk g)(zk )
(ε1 ,···,ε )∈Σβ
s k
1
q deg β+(S g)(zk )
sk
1
q (mk −) deg β+(Smk − g)(Tβ zk )
.
We estimate the inner summation.
s k
1
(ε1 ,···,ε )∈Σβ
q deg β+(S g)(zk )
≥
s +δ
1
(ε1 ,···,ε )∈Σβ
≥
q deg β+(S g)(zk )
1 1 q δ q (deg β+g)δ
(ε1 ,···,ε )∈Σβ
1 q deg β+(S g)(z )
s ,
C. Ma, S. Wang / Finite Fields and Their Applications 34 (2015) 176–191
189
where the first term q1δ appears when we switch zk ∈ Imk (ε1 , · · · , εmk ) to z ∈ I (ε1 , · · · , ε ). Hence by the definition of s , we have
s k
1
(ε1 ,···,ε )∈Σβ
q deg β+(S g)(zk )
≥
1 1 . q δ q (deg β+g)δ
Therefore,
sk
1
mk −
(ε+1 ,···,εmk )∈Σβ
q (mk −) deg β+(Smk − g)(Tβ zk )
≤ q δ(deg β+1+g) .
As a consequence, we get μ(In (x)) ≤ q δ(deg β+1+g) q −(nk−1 +tk−1 )(s0 −δ) deg β
1 ≤ |In (x)|s0 −(deg β+2+g)δ . q sk deg β
(II). When nk ≤ n ≤ nk + tk , note that Jk ⊂ In (x) we have μ(In (x)) = μ(Jk ) ≤ |Jk |s0 −δ ≤ |In (x)|s0 −δ . • Estimation for general balls. Write s0 = s0 − (deg β + 2 + g)δ. For any x ∈ C∞ and r > 0, let n be the integer such that |In+1 (x)| ≤ r < |In (x)|. Thus it follows that In+1 (x) ⊂ B(x, r) ⊂ In (x). As a consequence, μ(B(x, r)) ≤ μ(In (x)) ≤ |In (x)|s0 ≤ q deg β |In+1 (x)|s0 ≤ q deg β rs0 . Then by an application of the mass distribution principle (Proposition 3.2), we arrive at the final result. 4. Applications When f (x) = α deg β being a constant function, Theorem 1.2 leads to Corollary 4.1. dimH x ∈ I : Tβn x − y0 < q −nα deg β =
1 . 1+α
It should be noticed, when f is a constant function, the definition of the integer sequence {tk }k≥1 will no longer depend on {zk } (see formula (3.7)). Thus {tk } is uniform for all x ∈ C∞ . In other words, the dimension of the Cantor set C∞ depends only on
190
C. Ma, S. Wang / Finite Fields and Their Applications 34 (2015) 176–191
the behavior of f along a subsequence. Thus, it is easy to extend above result to general function by just letting {nk } be well chosen. Corollary 4.2. (See [6].) Let ψ be a positive function defined on natural numbers. Then we have dimH x ∈ I : Tβn x − y0 < q −ψ(n) =
1 , 1+α
where α = lim inf n→∞
ψ(n) . n deg β
If we take y0 = 0, then T n x − y0 = x − (
εn (x) ε1 (x) + ··· + ) := Wn (x). β βn
The quantify Wn (x) reflects how well x can be approximated by its convergent n εj (x) j=1 β j , which was investigated in detail in [28]. By Theorem 1.2, it follows directly that: Corollary 4.3. (See [28].) Let ψ be a positive function defined on natural numbers. Then we have dimH x ∈ I : Wn (x) < q −ψ(n) =
1 , 1+α
where α = lim inf n→∞
ψ(n) . n deg β
Acknowledgments This work is supported by the Science and Technology Development Fund of Macau (No. 069/2011/A), Fundamental Research Funds for the Central Universities 2014ZM0078 and NSFC (Nos. 11201155 and 11371148). References [1] Y. Bugeaud, S. Harrap, S. Kristensen, S. Velani, On shrinking targets for Zm actions on tori, Mathematika 56 (2) (2010) 193–202. [2] N. Chernov, D. Kleinbock, Dynamical Borel–Cantelli lemmas for Gibbs measures, Isr. J. Math. 122 (2001) 1–27. [3] K.J. Falconer, Fractal Geometry, Mathematical Foundations and Application, John Wiley, 1990.
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[4] A.-H. Fan, J. Schmeling, S. Troubetzkoy, A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation, Proc. Lond. Math. Soc. (3) 107 (5) (2013) 1173–1219. [5] A.H. Fan, J. Wu, A note on inhomogeneous Diophantine approximation with a general error function, Glasg. Math. J. 48 (2) (2006) 187–191. [6] Q.-B. Fan, S.-L. Wang, L.-P. Zhang, Recurrence of β-expansion over formal Laurent series, Monatshefte Math. 166 (3–4) (2012) 379–394. [7] J.L. Fernández, M.V. Melián, D. Pestana, Quantitative recurrence properties in expanding systems, arXiv:math/0703222, 2007. [8] S. Galatolo, D.H. Kim, The dynamical Borel–Cantelli lemma and the waiting time problems, Indag. Math. 18 (3) (2007) 421–434. [9] R. Ghorbel, M. Hbaib, S. Zouari, Purely periodic beta-expansions over Laurent series, Int. J. Algebra Comput. 22 (2) (2012) 1250010, 12 pp. [10] M. Hbaib, M. Mkaouar, Sur le bêta-développement de 1 dans le corps des séries formelles, Int. J. Number Theory 2 (3) (2006) 365–378. [11] R. Hill, S. Velani, The ergodic theory of shrinking targets, Invent. Math. 119 (1) (1995) 175–198. [12] M. Jellali, M. Mkaouar, K. Scheicher, Purely periodic β-expansions with Pisot unit bases over Laurent series, Int. J. Contemp. Math. Sci. 3 (5–8) (2008) 357–369. [13] B. Li, Y.-C. Chen, Chaotic and topological properties of β-transformations, J. Math. Anal. Appl. 383 (2011) 585–596. [14] B. Li, J. Wu, J. Xu, Metric properties and exceptional sets of β-expansions over formal Laurent series, Monatshefte Math. 155 (2) (2008) 145–160. [15] B. Li, B.-W. Wang, J. Wu, J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. 108 (2014) 159–186. [16] M. Lü, Diophantine approximation and beta-expansions over the field of formal Laurent series, J. Approx. Theory (2013), http://dx.doi.org/10.1016/j.jat.2013.06.008. [17] F. Maucourant, Dynamical Borel–Cantelli lemma for hyperbolic spaces, Isr. J. Math. 152 (2006) 143–155. [18] L.-M. Shen, B.-W. Wang, Shrinking target problems for beta-dynamical system, Sci. China Math. 56 (1) (2013) 91–104. [19] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung. 11 (1960) 401–416. [20] W. Philipp, Some metrical theorems in number theory, Pac. J. Math. 20 (1) (1967) 109–127. [21] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957) 477–493. [22] K. Scheicher, β-Expansions in algebraic function fields over finite fields, Finite Fields Appl. 13 (2007) 394–410. [23] K. Scheicher, J.M. Thuswaldner, Digit systems in polynomial rings over finite fields, Finite Fields Appl. 9 (3) (2003) 322–333. [24] B. Tan, B.-W. Wang, Quantitative recurrence properties for beta-dynamical system, Adv. Math. 228 (4) (2011) 2071–2097. [25] D. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Am. Math. Soc. 364 (10) (2012) 5395–5414. [26] J. Schmeling, S. Troubetzkoy, Inhomogeneous Diophantine approximation and angular recurrence for polygonal billiards, Mat. Sb. 194 (2) (2003) 129–144 (in Russian); translation in Sb. Math. 194 (1–2) (2003) 295–309. [27] P. Walters, An Introduction to Ergodic Theory, Grad. Texts Math., vol. 79, Springer, 1982. [28] S.-L. Wang, Approximation orders of formal Laurent series by β-expansions, Finite Fields Appl. 14 (2008) 1039–1055.
Contents lists available at ScienceDirect
Finite Fields and Their Applications www.elsevier.com/locate/ffa
Dynamical Diophantine approximation of beta expansions of formal Laurent series Chao Ma a , Shuailing Wang b,∗ a
Faculty of Information Technology and Department of General Education, Macau University of Science and Technology, Macau b Department of Mathematics, South China University of Technology, Guangzhou, 510640, PR China
a r t i c l e
i n f o
Article history: Received 3 January 2015 Received in revised form 30 January 2015 Accepted 31 January 2015 Available online xxxx Communicated by Arne Winterhof MSC: 11K55 28A80 Keywords: Dynamical Diophantine approximation β-Expansion Formal Laurent series Hausdorff dimension
a b s t r a c t Let Fq be a finite field with q elements and Fq ((X −1 )) be the field of the formal Laurent series with an indeterminant X. Let Tβ be the β-transformation defined on Fq ((X −1 )). This paper is concerned with the size of the following dynamically defined limsup set n−1 x ∈ Fq ((X −1 )) : Tβn x − y0 < q −(f (x)+···+f (Tβ x)) , for infinitely many n ∈ N , for any given y0 ∈ Fq ((X −1 )), where f is a positive continuous function and · is the norm, defined on Fq ((X −1 )). We show that its Hausdorff dimension is given as the solution s0 to the pressure function equation P(−s(deg β + f ), Tβ ) = 0. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (C. Ma), [email protected] (S. Wang). http://dx.doi.org/10.1016/j.ffa.2015.01.011 1071-5797/© 2015 Elsevier Inc. All rights reserved.
C. Ma, S. Wang / Finite Fields and Their Applications 34 (2015) 176–191
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1. Introduction Let (X, d) be a metric space and T : X → X be a transformation. Suppose that X is equipped with a T -invariant Borel probability measure μ. If T is further assumed to be ergodic with respect to the measure μ, then for any ball B in X of positive measure, the set x ∈ X : T n x ∈ B, for infinitely many n ∈ N
has full μ-measure. Chernov and Kleinbock [2] investigated the case when the fixed set B is replaced by a sequence of measurable sets {Bn }n≥1 in a general setting. More precisely, they concerned the size from the sense of the measure of the set (1.1) x ∈ X : T n x ∈ Bn , for infinitely many n ∈ N . This is called the dynamical Borel–Cantelli lemma [2] (see also [8,17]). Because of its analogy to the classic Diophantine approximation, this dynamically defined limsup set is also termed as the dynamical Diophantine approximation [4] (see also [5,26]). The set (1.1) is also called the shrinking target problem [11] and many mathematicians studied the size in the sense of the dimension of the set (1.1), see [1,15,18,24]. In this note, we consider the size of the set defined in (1.1) for β-expansions in the field of formal Laurent series, when Bn is a sequence of balls with given center. Let us first recall the field of formal Laurent series Fq ((X −1 )) and the β-expansion defined on it. Let Fq be a finite field with q elements, and Fq [X] be the ring of polynomials with coefficients in Fq and Fq (X) be the field of fractions. Then Fq ((X −1 )) =
+∞
xn X −n : xn ∈ Fq and n0 ∈ Z .
n=n0
The norm of x is defined to be x = q deg(x) , where deg(x) = − inf{n ∈ Z : xn = 0}, called the degree of x and deg(0) = −∞ with the convention. It is well known that · is a non-Archimedean norm on the field Fq ((X −1 )), that is, (1) (2) (3) (4)
x ≥ 0 with x = 0 if and only if x = 0; xy = x · y; αx + βy ≤ max(x, y) (∀α, β ∈ F); For all α, β ∈ F, α = 0, β = 0, if x = y, then αx + βy = max(x, y).
This norm induces a metric ρ as ρ(x, y) = x − y satisfying: (i) every point of a ball may be considered as the center of the ball; and (ii) if two balls intersect, the one with
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larger radius must contain the other. Moreover, Fq ((X −1 )) is a complete metric space under the metric ρ. Denote by I = x ∈ F((X −1 )) : x < 1 the unit ball of Fq ((X −1 )). The set I is isomorphic to n≥1 Fq and is an abelian compact group. As a consequence, there exists a unique normalized Haar measure μ on I given by
μ B(a, q −r ) = q −r , where B(a, q −r ) = x ∈ Fq ((X −1 )) : x − a < q −r is the disc of center a ∈ Fq ((X −1 )) and radius q −r with r ∈ Z. Note that μ(I) = 1 and (I, B (I), μ) is a probability space, where B (I) is a Borel field on I. Every x ∈ Fq ((X −1 )) has a unique (Artin) decomposition as x = [x] + {x}, where the integral part [x] of x belongs to Fq [X] and the fractional part {x} of x belongs to I. Let β ∈ Fq ((X −1 )) with β > 1. The β-transformation Tβ on I is defined as Tβ x = βx − [βx]. Then every x ∈ I can be represented by x=
ε1 (x) ε2 (x) εn (x) + + ··· + + ··· β β2 βn
(1.2)
with ε1 (x) = [βx] and εn (x) = ε1 (Tβn−1 (x)) for all n ≥ 2, called β-digits of x. The formula (1.2) is called the β-expansion of x, which was introduced by Scheicher [22], Hbaib and Mkaouar [10] independently. Theorem 1.1. (See [22].) An infinite sequence (εn )n≥1 in Fq [X] is the β-expansion of x ∈ I if and only if εn < β for all n ≥ 1. Denote by Σβ = {ε ∈ Fq [X] : ε < β} the set of possible β-digits of all formal Laurent series and by Σnβ := {ε ∈ Fq [X] : ε < β}n the set of possible words of length n. It is known that #Σβ = β, that is, the number of possible β-digits is β. Li, Wu and Xu [14] proved that Tβ is invariant and ergodic with respect to the Haar measure μ, and showed some limit theorems for β-digits and the Hausdorff dimensions of some exceptional sets according to the limit theorem. Wang [28] considered the metric theorem and exceptional sets about the approximation order of the β-expansion, that is, the convergent speed of the series in formula (1.2). Jellali, Mkaouar and Scheicher [12] (see also [9,23]) characterized the series whose β-expansion is purely periodic when β is a Pisot unit. Fan, Wang and Zhang [6] calculated the Hausdorff dimensions of the quantitative recurrence and hitting sets of β-transformation Tβ on I. Recently Lü [16] studied the Diophantine approximation problem for Tβ . The β-expansion on I was introduced as the analogue of that on real line. The β-expansion of the real numbers was given by Rényi [21] in 1957 and was studied extensively since then (see [13,19,20,25] etc).
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The metric theory about the Haar measure μ of the set defined in (1.1) falls into the general setting of Fernández, Melián and Pestana [7] which indicates that μ
x ∈ I : Tβn x ∈ Bn , i.o. n ∈ N =1
if and only if
μ(Bn ) = ∞.
n≥1
Here i.o. is a brevity for infinitely often. While on the other hand, it is clear but not trivial that there exist points violating this law. In this case, Hausdorff dimension serves as a tool to classify the size of zero-measure sets. This note is devoted to determining the Hausdorff dimensions of the following zero-measure sets. Let f be a positive continuous function defined on I and y0 be some fixed point in I. Define the limsup set S(f ) as S(f ) := x ∈ I : Tβn x − y0 < q −(Sn f )(x) , i.o. n ∈ N where Sn f denotes the ergodic sum, i.e., (Sn f )(x) =
n−1 i=0
f (Tβi x). We show that:
Theorem 1.2. The Hausdorff dimension of the set S(f ) is given as the solution s0 to the pressure function equation P(−s(deg β + f ), Tβ ) = 0.
(1.3)
For the definition and properties of pressure function, see Chapter 9 in Walters [27]. Remark. From Proposition 2.4 below, we know that the pressure equation (1.3) has the unique solution. 2. Preliminary Definition 2.1. For any given block (ε1 , ε2 , . . . , εn ) ∈ Σnβ , In (ε1 , ε2 , . . . , εn ) = {x ∈ I : ε1 (x) = ε1 , ε2 (x) = ε2 , . . . , εn (x) = εn } is called an n-th cylinder of the β-expansion. We denote by In (x) the n-th cylinder containing x. The measures μ(In (ε1 , ε2 , . . . , εn )) are very regular for any (ε1 , ε2 , . . . , εn ) ∈ Σnβ from the following proposition.
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Proposition 2.2. (See [14].) For any cylinder In (ε1 , ε2 , · · · , εn ), we have In (ε1 , ε2 , · · · , εn ) = B
ε2 εn 1 ε1 + 2 + ··· + n, β β β βn
.
(2.1)
As a consequence, μ(In (ε1 , ε2 , · · · , εn )) = β−n . Proposition 2.3. The β-transformation Tβ is continuous on I. Proof. For any x, y ∈ I, note that Tβ y −Tβ x = β ·y −x, then the result holds. 2 The topological pressure of a continuous map can be defined in different equivalent ways such as open covering, spanning set and separated set (see [27]). Since {I1 (ε) : ε ∈ Σβ } is a generator for Tβ , for any potential function ϕ ∈ C(Fq ((X −1 )), R), the topological pressure can be defined as P(ϕ, Tβ ) = lim
n→∞
1 logq n
q (Sn ϕ)(x) ,
(2.2)
(ε1 ,...,εn )∈Σn β
where x ∈ In (ε1 , . . . , εn ) and the limit does not depend on the choice of x. From (2.2) and the positiveness of f , we have the following. Proposition 2.4. The pressure function P(−s(deg β + f ), Tβ ) is strictly decreasing on s. 3. Proof of Theorem 1.2 We will prove Theorem 1.2 in this section. Naturally, the proof is divided into two parts by the estimations from above and below. Given s > 0 and a subset E of I, the s-Hausdorff measure is defined as Hs (E) = lim
⎧ ⎨ inf
δ→0 ⎩
j
|Bj |s
⎫ ⎬ ⎭
,
where the infimum is taken over all covers of E by balls Bj with diameter at most δ, and | · | denotes the diameter of a set. The Hausdorff dimension of the set E is given by dimH (E) = inf{s ≥ 0 : Hs (E) = 0}. 3.1. Upper bound The upper bound of dimH S(f ) can be obtained by considering the natural covering system. Evidently,
C. Ma, S. Wang / Finite Fields and Their Applications 34 (2015) 176–191
S(f ) =
∞ ∞
J(ε1 , · · · , εn ),
181
(3.1)
N =1 n=N (ε1 ,···,εn )∈Σn β
where J(ε1 , · · · , εn ) = x ∈ In (ε1 , · · · , εn ) : Tβn x − y0 < q −(Sn f )(x) . Then, for any N ≥ 1, {J(ε1 , · · · , εn ) : (ε1 , · · · , εn ) ∈ Σnβ , n ≥ N } is a natural cover of the set S(f ). Note that sup {(Sn f )(x) − (Sn f )(y) : In (x) = In (y)} ≤
n
sup {|f (x) − f (y)| : Ii (x) = Ii (y)} ,
i=1
since the continuity of f implies n
sup {|f (x) − f (y)| : Ii (x) = Ii (y)} = o(n);
i=1
for any δ > 0, there exists an integer N ∈ N such that for all n ≥ N , sup (Sn f )(x) − (Sn f )(y) : In (x) = In (y) ≤ nδ. Therefore, if we replace x by zn :=
εn (x) ε1 (x) + ··· + ∈ In (x) β βn
in (Sn f )(x), there is a loss at most nδ, that is, |(Sn f )(zn ) − (Sn f )(x)| ≤ nδ.
(3.2)
J(ε1 , · · · , εn ) ⊂ x ∈ In (ε1 , · · · , εn ) : Tβn x − y0 < q −(Sn f )(zn )+nδ .
(3.3)
Thus, we have
While, it is clear that the set in the right side of (3.3) is the ball B
εn y0 −n deg β −(Sn f )(zn )+nδ ε1 . + ··· + n + n,q ·q β β β
Hence we obtain, for large n, that J(ε1 , · · · , εn ) ≤ q −n deg β−(Sn f )(zn )+nδ .
(3.4)
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182
Combining (3.1) and (3.4), for any s > s0 , we have Hs (S(f )) ≤ lim inf N →∞
≤ lim inf N →∞
∞
J(ε1 , · · · , εn )s
n=N (ε1 ,···,εn )∈Σn β ∞
q (−n deg β−(Sn f )(zn )+nδ)s .
n=N (ε1 ,···,εn )∈Σn β
By Proposition 2.4, −αs := P(−s(deg β + f ), Tβ ) < 0 since s > s0 . From (2.2), we know that when n is large enough,
q (−n deg β−(Sn f )(zn ))s ≤ q −
αs 2
n
.
(ε1 ,···,εn )∈Σn β
Therefore, H (S(f )) ≤ lim inf s
N →∞
∞
q −(
αs 2
−δs)n
,
n=N
which implies Hs (S(f )) < +∞ by choosing δ <
αs 2s .
Thus dimH S(f ) ≤ s0 .
3.2. Lower bound For any (ε1 , · · · , εn ) ∈ Σnβ , in need of picking a point zn in In (ε1 , · · · , εn ), we always take, with no extra announcement, zn =
εn ε1 + ··· + n. β β
Lemma 3.1. Let g be a positive continuous function and y0 ∈ I. Then the Hausdorff dimension s of the following set S(g) := x ∈ I : Tβn x − y0 < q −(Sn g)(zn ) with zn ∈ In (x), i.o. n ∈ N fulfills the pressure equation P(−s(deg β + g), Tβ ) = 0. We claim that Lemma 3.1 implies Theorem 1.2. Indeed, take g = f + (deg β + f )δ for any δ > 0. By the continuity of f , it is clear that S(g) ⊂ S(f ) since (Sn g)(zn ) ≥ (Sn f )(x) from (3.2). Let s0 be the solution of Eq. (1.3), then s0 (deg β + g), Tβ 0 = P(−s0 (deg β + f ), Tβ ) = P − 1+δ
C. Ma, S. Wang / Finite Fields and Their Applications 34 (2015) 176–191
by noting f =
g−δ deg β . 1+δ
183
As a consequence, dimH S(f ) ≥ dimH S(g) =
s0 , 1+δ
which gives a good lower bound of the Hausdorff dimension of the set S(f ) by letting δ → 0. Therefore, the left is to prove Lemma 3.1. The following mass distribution principle will be applied to obtain the lower bound of dimH S(g). Proposition 3.2. (See Falconer [3].) Let E be a Borel measurable set in a metric space X and μ be a Borel measure with μ(E) > 0. Assume that there exist two positive constants c, η such that, for any set U with diameter diam(U ) less than η, μ(U ) ≤ c diam(U )s , then dimH E ≥ s. Directly by this proposition, we are led to the following three steps: firstly, construct a Cantor subset C∞ of S(g); secondly, define a measure μ supported on C∞ , and at last, estimate the Hölder exponent of the measure μ. 3.2.1. Construction of a Cantor set First we outline the idea for the construction of a Cantor subset of S(g). Main strategy: By the algorithm of β-expansion, we know that every x ∈ I has a unique symbolic representation by the sequence of its digits (ε1 (x), ε2 (x), · · ·). Such a representation makes it much convenient to achieve that two points x, y are close enough just by letting x, y have the same prefix for a long time in their symbolic representations. More precisely, to make sure that x ∈ B(y, r) for some r > 0, we choose the integer t such that |It (y)| ≤ r < |It−1 (y)| and x can be chosen as any element in It (y). For a digit block (ε1 , · · · , εn ) ∈ Σnβ , define tn = tn (ε1 , · · · , εn ) to be the integer such that |Itn (y0 )| ≤ q −(Sn g)(zn ) < |Itn −1 (y0 )|,
(3.5)
n ε where zn = j=1 βjj . Now we construct a Cantor subset of S(g). In the sequel, we denote by (b1 , b2 , · · · , ) the sequence of the digits of y0 in its β-expansion. Choose a largely sparse sequence {mk }k≥1 with m1 1,
k = o(1), m1 + · · · + mk−1 mk (k ≥ 1). mk
(3.6)
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(1)
(1)
1 Level 1 of the Cantor set. Take n1 = m1 . For each (ε1 , · · · , εm1 ) ∈ Σm β , for simplicity of notation, we denote by t1 the integer defined in (3.5), i.e. t1 is the integer satisfying
q −t1 deg β ≤ q −(Sn1 g)(z1 ) < q −(t1 −1) deg β , (1)
where z1 =
ε1 β
+ ··· +
ε(1) n1 β n1
(1)
(1)
∈ In1 (ε1 , · · · , εn1 ). (1)
(1)
Note that the integer t1 depends on the sequence (ε1 , · · · , εm1 ), but we omit this dependence in the notation for simplicity. Then the first level of the Cantor set is defined as
(1) (1) m1 (1) C (1) = In1 +t1 ε1 , · · · , ε(1) , b , · · · , b , · · · , ε ) ∈ Σ : (ε . t1 m1 1 m1 1 β Other levels of the Cantor set are defined by an induction. Level k of the Cantor set. Assume that the (k − 1)-th level of the Cantor set has been well constructed. Now fix a cylinder Jk−1 = Ink−1 +tk−1 (E (k−1) ) ∈ C (k − 1) and write (k) (k) k nk = nk−1 + tk−1 + mk . For any (ε1 , · · · , εmk ) ∈ Σm β , determine tk by (3.5) for the digit block (k)
(E (k−1) , ε1 , · · · , ε(k) mk ), i.e. tk is the integer satisfying q −tk deg β ≤ q −(Snk g)(zk ) < q −(tk −1) deg β , (k)
(3.7)
(k)
where zk ∈ Ink (E (k−1) , ε1 , · · · , εmk ). The right inequality in (3.7) indicates that tk is bounded from above by nk , more precisely, tk < 1 +
nk g , deg β
(3.8)
where g is the maximal norm of g in C(I, R). Then a subcollection of the second level is given as
(k) (k) mk (k) C (k, Jk−1 ) = Ink +tk E (k−1) , ε1 , · · · , ε(k) . mk , b1 , · · · , btk : (ε1 , · · · , εmk ) ∈ Σβ Then we define the k-th level as C (k) = C (k, Jk−1 ) : Jk−1 ∈ C (k − 1) . The Cantor set. Continuing this process, we obtain a nested sequence {C (k)}k≥1 composed of cylinders, called basic cylinders. And then the desired Cantor set is C∞ =
∞
k=1 Jk ∈C (k)
Jk .
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From the construction of C∞ , we have C∞ ⊂ S(g).
3.2.2. Supporting measure We distribute a probability measure μ on C∞ . For each k ≥ 1, define sk as the solution to the equation
1=
q −s(mk deg β+(Smk g)(zk )) ,
(3.9)
mk
(ε1 ,···,εmk )∈Σβ
k where zk is chosen in Imk (ε1 , · · · , εmk ) for each (ε1 , · · · , εmk ) ∈ Σm β . Comparing the elements zk and zk in (3.7) and (3.9), it should be noticed that
(1)
(k)
(k) zk ∈ Ink (ε1 , · · · , ε(1) m1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk ),
while zk ∈ Imk (ε1 , · · · , ε(k) mk ). (k)
Now we collect some simple observations from these relations which will be used later. One is the following n
Tβ k−1
+tk−1
zk = zk , |(Snk g)(zk ) − (Smk g)(zk )| ≤ mk δ,
(3.10)
where the inequality holds because of (3.6) and (3.8). It is clear that, by the definition of the pressure function, we have Lemma 3.3. lim sk = s0 .
(3.11)
k→∞
For any basic cylinder Jk ∈ C (k), let Jk−1 ∈ C (k − 1) be its mother cylinder, i.e. Jk ∈ C (k, Jk−1 ). Then by the construction of C (k, Jk−1 ), the cylinders Jk and Jk−1 can be expressed as (k)
(k−1) Jk = Ink +tk (E (k−1) , ε1 , · · · , ε(k) ). mk , b1 , . . . , btk ), Jk−1 = Ink−1 +tk−1 (E
The measure of Jk is defined as μ(Jk ) = μ(Jk−1 )
s k
1
q mk deg β+(Smk g)(zk )
where zj ∈ Imj (ε1 , · · · , εmj ), for 1 ≤ j ≤ k. (j)
(j)
=
k
s j
1
j=1
q mj deg β+(Smj g)(zj )
, (3.12)
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Then we extend the definition of the set function μ to be a measure supported on C∞ . For each x ∈ C∞ , by the construction of C∞ , there exist a sequence of digit blocks (k) (k) {ε1 , · · · , εmk }k≥1 and a sequence of integers {tk }k≥1 such that (1)
(k)
(k) x = [ε1 , · · · , ε(1) m1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk , b1 , · · · , btk , · · ·].
Moreover, the integer sequence {tk }k≥1 satisfies (3.7), i.e., q −tk deg β ≤ q −(Snk g)(zk ) < q −(tk −1) deg β , (1)
(1)
(k)
(3.13)
(k)
where zk ∈ Ink (ε1 , · · · , εm1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk ). For each k ≥ 1 and nk−1 + tk−1 < n ≤ nk + tk , we define the measure of In (x) in the following way. (1). When nk−1 + tk−1 < n < nk . Write n = nk−1 + tk−1 + . Then the cylinder In (x) can be expressed as (k)
(k)
In (x) = In (E (k−1) , ε1 , · · · , ε ). Then we define
μ(In (x)) = (k)
(k)
(k)
(k)
(k)
μ(Ink +tk (E (k−1) , ε1 , · · · , ε , ε+1 , · · · , ε(k) mk , b1 , · · · , btk )) mk −
(ε+1 ,···,εmk )∈Σβ
where the summation is taken over all digit blocks (ε+1 , · · · , εmk ) ∈ Σβmk − . Thus, (k)
μ(In (x)) k−1 = j=1
s j
1
q mj deg β+(Smj g)(zj )
(k)
sk
1
m − (k) (k) (ε+1 ,···,εmk )∈Σβ k
q mk deg β+(Smk g)(zk )
. (3.14)
(2). When nk ≤ n ≤ nk + tk ,
μ(In (x)) = μ(Jk ) =
k
s j
1
j=1
q mj deg β+(Smj g)(zj )
.
(3.15)
The cylinders that do not intersect C∞ are allocated zero by the distribution μ. Thus the measure μ is well defined on all cylinders. By Kolmogorov’s extension theorem, the measure μ can be defined on (I, B(I)). It is easy to know that the support of μ is C∞ from the construction of μ.
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3.2.3. Hölder exponent of the measure • Estimation of the μ-measure of cylinders. Of course, we only consider the cylinders with non-zero μ-measure. Given a basic cylinder Jk ∈ C (k), write it as (1)
(k)
(k) Jk = Ink +tk (ε1 , · · · , ε(1) m1 , b1 , · · · , bt1 , · · · , ε1 , · · · , εmk , b1 , . . . , btk ).
At first, we estimate the length of Jk . It is clear that |Jk | = q −(nk +tk ) deg β = q −(m1 +···+mk ) deg β q −(t1 +···+tk ) deg β . Following from (3.13) on the definition of tk , we have q −k deg β
k
q −(mj deg β+(Snj g)(zj )) ≤ |Jk | ≤
j=1
k
q −(mj deg β+(Snj g)(zj )) .
(3.16)
j=1
Recalling the measure of Jk (see formula (3.12)) and the difference on zj and zj for 1 ≤ j ≤ k (see formula (3.10)), it follows directly that μ(Jk ) ≤ |Jk |s0 −δ .
(3.17)
For any x ∈ C∞ , we will check the Hölder exponent of μ along the cylinders containing x. For each n ≥ n1 + t1 , let k > 1 be the integer such that nk−1 + tk−1 < n ≤ nk + tk . (I). When nk−1 + tk−1 < n < nk , writing n = nk−1 + tk−1 + ,
μ(In (x)) = μ(Jk−1 )
m − (k) (k) (ε+1 ,···,εmk )∈Σβ k
= μ(Jk−1 )
1
s k
1
q mk deg β+(Smk g)(zk )
s k
1
q deg β
≤ μ(Jk−1 )
s k
1
m − (k) (k) (ε+1 ,···,εmk )∈Σβ k
s k
q (mk −) deg β+(Smk g)(zk )
sk
1
q deg β
mk −
(ε+1 ,···,εmk )∈Σβ
q (mk −) deg β+(Smk − g)(Tβ zk )
where the last inequality follows by the simple fact of positiveness of g. Now we estimate the last summation in detail. Recall the definition of sk . 1=
s k
1
m (ε1 ,···,ε ,ε+1 ,···,εmk )∈Σβ k
q mk deg β+(Smk g)(zk )
,
k where zk ∈ Imk (ε1 , · · · , εmk ) for each (ε1 , · · · , εmk ) ∈ Σm β . By Lemma 3.3, there exists 0 ∈ N such that for any , k > 0 , |s − sk | < δ.
,
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When ≤ 0 ,
1≥
q mk deg β+(Smk g)(zk )
mk −
(ε+1 ,···,εmk )∈Σβ
≥
s k
1
1
sk
1
.
q (deg β+g)
q (mk −) deg β+(Smk − g)(Tβ zk )
mk −
(ε+1 ,···,εmk )∈Σβ
Hence we have
sk
1
≤ q (deg β+g)
q (mk −) deg β+(Smk − g)(Tβ zk )
m − (ε+1 ,···,εmk )∈Σβ k
≤ q 0 (deg β+g) := C1 . So, together with the first inequality in (3.17), we get
s k
1
μ(In (x)) ≤ C1 μ(Jk−1 )
s0 −δ
1
≤ C1 q deg β q (nk−1 +tk−1 ) deg β s0 −δ 1 ≤ C1 = C1 |In (x)|s0 −δ . n deg β q
1
s k
q deg β
When > 0 , |s − sk | < δ, so,
1≥
s k
1
mk
(ε1 ,···,ε ,ε+1 ,···,εmk )∈Σβ
=
mk −
(ε+1 ,···,εmk )∈Σβ
×
q mk deg β+(Smk g)(zk )
(ε1 ,···,ε )∈Σβ
s k
1
q deg β+(S g)(zk )
sk
1
q (mk −) deg β+(Smk − g)(Tβ zk )
.
We estimate the inner summation.
s k
1
(ε1 ,···,ε )∈Σβ
q deg β+(S g)(zk )
≥
s +δ
1
(ε1 ,···,ε )∈Σβ
≥
q deg β+(S g)(zk )
1 1 q δ q (deg β+g)δ
(ε1 ,···,ε )∈Σβ
1 q deg β+(S g)(z )
s ,
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189
where the first term q1δ appears when we switch zk ∈ Imk (ε1 , · · · , εmk ) to z ∈ I (ε1 , · · · , ε ). Hence by the definition of s , we have
s k
1
(ε1 ,···,ε )∈Σβ
q deg β+(S g)(zk )
≥
1 1 . q δ q (deg β+g)δ
Therefore,
sk
1
mk −
(ε+1 ,···,εmk )∈Σβ
q (mk −) deg β+(Smk − g)(Tβ zk )
≤ q δ(deg β+1+g) .
As a consequence, we get μ(In (x)) ≤ q δ(deg β+1+g) q −(nk−1 +tk−1 )(s0 −δ) deg β
1 ≤ |In (x)|s0 −(deg β+2+g)δ . q sk deg β
(II). When nk ≤ n ≤ nk + tk , note that Jk ⊂ In (x) we have μ(In (x)) = μ(Jk ) ≤ |Jk |s0 −δ ≤ |In (x)|s0 −δ . • Estimation for general balls. Write s0 = s0 − (deg β + 2 + g)δ. For any x ∈ C∞ and r > 0, let n be the integer such that |In+1 (x)| ≤ r < |In (x)|. Thus it follows that In+1 (x) ⊂ B(x, r) ⊂ In (x). As a consequence, μ(B(x, r)) ≤ μ(In (x)) ≤ |In (x)|s0 ≤ q deg β |In+1 (x)|s0 ≤ q deg β rs0 . Then by an application of the mass distribution principle (Proposition 3.2), we arrive at the final result. 4. Applications When f (x) = α deg β being a constant function, Theorem 1.2 leads to Corollary 4.1. dimH x ∈ I : Tβn x − y0 < q −nα deg β =
1 . 1+α
It should be noticed, when f is a constant function, the definition of the integer sequence {tk }k≥1 will no longer depend on {zk } (see formula (3.7)). Thus {tk } is uniform for all x ∈ C∞ . In other words, the dimension of the Cantor set C∞ depends only on
190
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the behavior of f along a subsequence. Thus, it is easy to extend above result to general function by just letting {nk } be well chosen. Corollary 4.2. (See [6].) Let ψ be a positive function defined on natural numbers. Then we have dimH x ∈ I : Tβn x − y0 < q −ψ(n) =
1 , 1+α
where α = lim inf n→∞
ψ(n) . n deg β
If we take y0 = 0, then T n x − y0 = x − (
εn (x) ε1 (x) + ··· + ) := Wn (x). β βn
The quantify Wn (x) reflects how well x can be approximated by its convergent n εj (x) j=1 β j , which was investigated in detail in [28]. By Theorem 1.2, it follows directly that: Corollary 4.3. (See [28].) Let ψ be a positive function defined on natural numbers. Then we have dimH x ∈ I : Wn (x) < q −ψ(n) =
1 , 1+α
where α = lim inf n→∞
ψ(n) . n deg β
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