Exceptional parameters of linear mod one transformations and fractional parts {ξ(p/q) n }

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1 Excetional arameters of linear mod one transformations and fractional arts {ξ(/) n } DoYong Kwon, Deartment of Mathematics, Chonnam National University, Gwangju , Reublic of Korea Received *****; acceted after revision Presented by Abstract We study excetional arameters of linear mod one transformations. The resent note roves that the set of such values has Hausdorff dimension zero. This answers the uestion osed by Bugeaud. To cite this article: D.Y. Kwon, C. R. Acad. Sci. Paris, Ser. I??? (201?). Résumé Paramètres excetionnels de transformations linéaires mod un et arties fractionnaires {ξ(/) n } Nous étudions aramètres excetionnels de transformations linéaires mod un. La résente note rouve ue l ensemble de ces valeurs a Hausdorff dimension zéro. Cela réond à la uestion osée ar Bugeaud. Pour citer cet article : D.Y. Kwon, C. R. Acad. Sci. Paris, Ser. I??? (201?). 1. Introduction Let { } denote the fractional art. For an interval [s, s + t) [0, 1), and corime integers, with > 2, we set Z / (s, s + t) := {ξ > 0 : s {ξ(/) n } < s + t for all integers n 0}. A well-known connection between Waring s roblem and the fractional arts {(3/2) n } leaded Mahler, in [13], to consider a real number ξ Z 3/2 (0, 1/2). Mahler called such a hyothetical number a Z-number. He roved that there are at most countably many Z-numbers, but was unable to determine whether Z 3/2 (0, 1/2) is emty or not. A series of researches since then tell us that, let alone Mahler s original Z-number roblem, it is nontrivial enough to determine whether Z / (s, s + 1 ) is emty or not [5,1,3]. address: doyong@jnu.ac.kr (DoYong Kwon). Prerint submitted to the Académie des sciences Setember 17, 2014

2 The resent note is another small ste in this direction. It is worthwhile to mention here that if = 1 then the set Z /1 (s, s + 1 ) is now comletely understood thanks to Bugeaud and Dubickas [2]. Let τ [0, 1) be real. For any k Z, ut ε k (τ) := kτ (k 1)τ, where denotes the integral art. Proosition 1.1 ([1]) Let > 2 be corime integers. Then the set Z / (s, s + 1 ) is emty if there exists a reduced rational a/b with b > a 1 such that ) k ( ) b k=1 ε k(a/b) + ) k ( ) b 1 k=1 ε k(a/b) + ( ( ) b 1 {( )s} ( ( ) b 1. And the set of s satisfying the ineuality is of full Lebesgue measure in [0, 1 1 ]. Conversely, if Z /(s, s+ 1 ) is nonemty for some s [0, 1 1 ], then there exists an irrational τ (0, 1) such that {( )s} = ( ) k ε k (τ). Later, in [3], Dubickas showed that the set Z / (s, s + 1 ) is actually emty for every s [0, 1 1 ] rovided and satisfy 1 < < < 2. In articular, the set Z 3/2 (s, s ) is emty for each s [0, 2 3 ]. In the case of > 2, the determination of whether or not the set Z / (s, s + 1 ) is emty still remains oen. Bugeaud obtained the above roosition by a careful study of linear mod one transformations. For real numbers β > 1 and 0 α < 1, the linear mod one transformation f β,α is defined by Via setting k=1 f β,α (x) := {βx + α} for x [0, 1). S β,α := {x [0, 1) : 0 f n β,α(x) < 1/β for all n 0}, the linear mod one transformation enters the icture as the next roosition says. Proosition 1.2 ([5]) Let > 2 be corime integers and s [0, 1 1 ]. If S /,{( )s} is a finite set, then the set Z / (s, s + 1 ) is emty. Therefore, we have a good reason to secify the following set: for a fixed β > 1, E β := {α [0, 1) : S β,α is an infinite set}. Flatto et al. [5] susected that E β has Lebesgue measure zero and is a nonemty erfect set. But the erfectness of E β turns out to be false. For β > 1, ut γ = 1/β. We define intervals Jb a(γ) by J 1 1 (γ) := [γ, 1), and by [ Jb a k=1 (γ) := ε k(a/b)γ k + γ b ] k=1 1 + γ + + γ b 1, ε k(a/b)γ k + γ b γ + + γ b 1, for corime integers b > a 1. One observes that if β = / then the interval Jb a (γ) is given by the ineuality in Proosition 1.1. b 1 1. In [1, Lemma3], k=1 was used instead of k=1, which, however, makes no difference because ε (b 1)(a/b) = 0. We adot here to make the notation coherent with Proosition 1.1. k=1 2

3 Proosition 1.3 ([1]) For any real β > 1, the set E β has Lebesgue measure zero, is uncountable, and is not closed. More recisely, E β = [0, 1) \ Jb a (γ). 1 a b gcd(a,b)=1 Owing to this result, one notes that the set Z / (s, s + 1 ) is emty, ossibly excet when s lies in a Lebesgue negligible set. In [1, Remark 3], Bugeaud osed a roblem to determine the Hausdorff dimension of the set E β. The resent note settles this roblem. Main Theorem. The set E β has Hausdorff dimension zero. This comutation is ossible by recognizing the unexected connection with ower series whose coefficient are Sturmian words, which have been investigated by the author [7,8,10,11]. Note that, in the context of a certain generalization of [2], this tye of ower series aear as well [9]. 2. Preliminaries on Sturmian words Let Z (res. N) be the set of integers (res. nonnegative integers). For α, ρ [0, 1], two arithmetic functions s α,ρ, s α,ρ : Z A := {0, 1} are defined by s α,ρ (n) := α(n + 1) + ρ αn + ρ, s α,ρ(n) := α(n + 1) + ρ αn + ρ, where denotes the ceiling function. Then two (right) infinite words s α,ρ := s α,ρ (0)s α,ρ (1) and s α,ρ := s α,ρ(0)s α,ρ(1) are termed lower and uer mechanical words, resectively, with sloe α and intercet ρ. The bi-infinite mechanical words are also considered and written as s α,ρ := s α,ρ ( 1)s α,ρ (0)s α,ρ (1), s α,ρ := s α,ρ( 1)s α,ρ(0)s α,ρ(1), where the zeroth letters are marked with underlines. If α is irrational, then s α,ρ and s α,ρ are called Sturmian words. The case of ρ = 0 is of our secial interest, which is stated as a lemma below. For any finite word u A, we write ũ for its reversal, and a word u satisfying ũ = u is said to be a alindrome. We mean by u ω := uuu (res. ω u := uuu) the right (res. left) infinite word infinitely concatenated by u. If s A N is a right infinite word, then s is the left infinite word that is defined in a natural manner. Lemma 2.1 Let α [0, 1] be real. (a) If α is irrational, then there exists a right infinite word c α A N such that s α,0 = c α 10c α, and s α,0 = c α 01c α. (b) If α = a/b is rational with gcd(a, b) = 1, then there exists a alindrome z a,b A of length b 2 such that s α,0 = ω (0z a,b 1)0(z a,b 10) ω, and s α,0 = ω (1z a,b 0)1(z a,b 01) ω. PROOF. See [12]. Remark 1 (i) In the lemma, c α and z a,b are called the characteristic word and the central word resectively in the literature. 3

4 (ii) If b = 2, then z a,b is the emty word. In case b = 1, we adot convention that s 0,0 = s 0,0 = ω 000 ω and s 1,0 = s 1,0 = ω 111 ω. For β > 1, let ( ) β send each infinite word a 0 a 1 A N to a real number i=0 a i/β i+1. Then a real function µ β : [0, 1] R is defined by µ β (x) := (s x,0) β = i=0 s x,0(i) β i+1. A detailed real analysis on µ β was ursued in [10,11]. However, the following results are enough for our urose. Proosition 2.2 For any fixed β > 1, the function µ β (x) is strictly increasing. If α [0, 1] is irrational, then the function µ β (x) is continuous at x = α. On the other hand, at rational α 0, µ β (x) is left-continuous but not right-continuous. PROOF. See [11, Lemma 3.1]. Proosition 2.3 Let β > 1 be fixed, and suose that α = a/b [0, 1] is rational with gcd(a, b) = 1. (a) The right limits of µ β (x) at rational oints are given by µ β (0+) = (1(0 ω )) β = 1 β, and by µ β (α+) = (1(z a,b 10) ω ) β. (b) µ β (0+) µ β (0) = 1 β, and µ β (α+) µ β (α) = (1(z a,b 10) ω ) β ((1z a,b 0) ω ) β = β(β b 1). (c) where β 1 β(β b 1) = 1 β 0 a/b<1 gcd(a,b)=1 when a/b = 0/1 = 0. β(β b 1) = 1 = µ β(1), PROOF. In [10], see Lemma 2.3 for (a), Theorem 2.4 for (b), and Theorem 3.1 for (c). Proosition 2.3 says that µ β is a ure jum distribution. Noting that { c τ 10c τ if τ is irrational, ε 1 (τ)ε 0 (τ)ε 1 (τ) = ω (10z a,b )1(0z a,b 1) ω if τ = a/b with gcd(a, b) = 1, Proosition 1.1 can be restated more neatly. Lemma 2.4 Let > 2 be corime integers and ut β = /. Then the set Z / (s, s + 1 ) is emty if there exists a reduced rational a/b with b > a 1 such that µ β (a/b) {( )s} + µ β((a/b)+). Conversely, if Z / (s, s + 1 ) is nonemty for some s [0, 1 1 ], then there exists an irrational τ (0, 1) such that {( )s} + = µ β(τ). 4

5 PROOF. Divided by, the ineuality in Proosition 1.1 becomes which is, in turn, followed by (0(z a,b 01) ω ) β µ β (a/b) = ((1z a,b 0) ω ) β {( )s} {( )s} (0(z a,b 10) ω ) β, + 1 β (1(z a,b10) ω ) β = µ β ((a/b)+). For the case where Z / (s, s + 1 ) is nonemty, a similar argument shows that {( )s} + 1 β = (1c τ ) β = µ β (τ). 3. Proof of Main Theorem Let Ẽβ := µ β ([0, 1] \ Q) be the image of µ β at irrational oints. Lemma 2.4 tells us that E β \ {0} is obtained by translation and then dilation of Ẽβ. More recisely, the same argument as in the roof of Lemma 2.4 derives Ẽ β = 1 E β + 1 β. So we comute the Hausdorff dimension of Ẽ β. Among diverse techniues for calculating Hausdorff dimensions is the following. Lemma 3.1 ([4]) Suose that a set S can be covered by n k sets of diameter at most δ k with lim k δ k = 0. Then log n k dim H S lim inf. k log δ k We recall the Farey seuence. The reader is referred to [6] for details. For ositive integer k, the Farey seuence F k of order k is the seuence of reduced fractions a/b [0, 1] with b k, arranged in increasing order. For examle, F 1 = {0/1, 1/1}, F 2 = {0/1, 1/2, 1/1}, F 3 = {0/1, 1/3, 1/2, 2/3, 1/1},.... The next lemma is one of folklores. Lemma 3.2 ([6]) The number of terms in F k fulfills k F k 1 = Φ(k) := where ϕ is the Euler totient function. We are now in a osition to rove our main theorem. ϕ(i) = 3k2 π 2 + O(k log k), Proof of Main Theorem. We construct a series of collections of sets that cover Ẽβ. For each integer k 1, let F k = {0 = r 1 < r 2 < < r Φ(k) < r Φ(k)+1 = 1} be the Farey seuence of order k. Then a set [0, 1 β 1 ) \ Φ(k) [µ β(r i ), µ β (r i +)] covers Ẽβ, and consists of n k := Φ(k) intervals. Suose r i = a i /b i. Then Proosition 2.3 shows that the diameter of each interval is less than or eual to δ k := 1 Φ(k) β(β bi 1) = 1 k β() 5 b=2 ϕ(b)() β(β b 1),

6 and also that δ k tends to zero as k aroaches infinity. Let us ick a number θ in the interval (1, β). We claim that δ k θ k for all sufficiently large k. Aealing to Proosition 2.3 again, one deduces that ϕ(b)() δ k = β(β b b 1 1) β β b 1 = b k() + 1 = β βb ()β k, b=k+1 b=k+1 where ϕ(b) b 1, and where β b 1 β b 1 as long as k log β (). Now the claim follows. We comute the Hausdorff dimension of Ẽβ: log n k log(3k 2 /π 2 + O(k log k)) dim H Ẽ β lim lim = 0. k log δ k k k log θ 1 That the diameter of each interval in [0, β 1 ) \ Φ(k) [µ β(r i ), µ β (r i +)] is less than or eual to δ k is a very rough estimate. Moreover, δ k tends to zero very uickly. Conseuently, the above roof urges us to introduce thinner measures than the Hausdorff measure. For any set U R n, we write U := su{ x y : x, y U}. Let h : R + R + be a continuous and increasing function. For any set S R n, let us define Hδ h (S) := inf{ h( U i ) : S U i, U i δ}. Then H h (S) := lim δ 0 H h δ (S) is known to be a measure. We have roved, in the above, that Hh (Ẽβ) = 0 whenever h(t) = t s with s > 0. On the other hand, if we set h(t) := ( log t) 2 ε for ε > 0, then H h is much thinner measure than any s-dimensional Hausdorff measure. Since H h δ k (Ẽβ) n k h(δ k ), one deduces b=k H h (Ẽβ) lim n 3k 2 /π 2 + O(k log k) kh(δ k ) lim k k (k log θ) 2+ε = 0. It seems to be a challenge to find a function h for which 0 < H h (Ẽβ) <. Such h is called an (exact) dimension function in the literature. See [4, Section 2.5]. Acknowledgements Referee s comments and suggestions are gratefully acknowledged, which enhanced readability and motivated the author to consider dimension functions. This research was suorted by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2013R1A1A ). References [1] Y. Bugeaud. Linear mod one transformations and the distribution of fractional arts {ξ(/) n }. Acta Arith. 114 (2004), no.4, [2] Y. Bugeaud and A. Dubickas. Fractional arts of owers and Sturmian words. C. R. Math. Acad. Sci. Paris 341 (2005), no. 2, [3] A. Dubickas. Powers of a rational number modulo 1 cannot lie in a small interval. Acta Arith. 137 (2009), no.3, [4] K. Falconer. Fractal geometry. Mathematical foundations and alications. Second edition. John Wiley & Sons, Inc., Hoboken, NJ,

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