Powers of a rational number modulo 1 cannot lie in a small interval

Abstract

Powers of a rational number modulo 1 cannot lie in a small interval by Artūras Dubickas (Vilnius)1. Introduction. Let throughout R, Z and N be the sets of real numbers, integers and positive integers, respectively. We will denote by [x] and {x} the integral part and the fractional part of x ∈ R, respectively. For an interval [s, s + t) ⊂ [0, 1) and two integers p, q, where 1 < q < p, putIn [14] Mahler asked whether the set Z 3/2 (0, 1/2) is empty or not. A hypothetical ξ ∈ Z 3/2 (0, 1/2) is called a Z-number. It seems very likely that Z-numbers do not exist. An important step towards solution of this problem has been made by Flatto, Lagarias and Pollington [12] (see also [11]). It was proved in [12] that for coprime positive integers p > q > 1 and any ξ = 0 the inequalityholds. A generalization of (1) to powers of algebraic numbers is given in [9]. The case of positive integers, namely, p ≥ 2, q = 1 was studied in [7]. Inequality (1) implies that the fractional parts {ξ(p/q) n }, n = 0, 1, 2, . . . , cannot lie in an interval of length strictly smaller than 1/p. Can they all lie in an interval of length 1/p? This small step towards Mahler's problem turns out to be very difficult. It was shown in [12] that the set of s ∈ [0, 1−1/p] for which Z p/q (s, s + 1/p) is empty is everywhere dense in [0, 1 − 1/p]. Naturally, it was conjectured that Z p/q (s, s + 1/p) is empty for each s ∈ [0, 1 − 1/p] (see p. 138 in [12]).This problem is still open, although Bugeaud has made some progress in this direction in [6]. He was able to prove that Z p/q (s, s + 1/p) is empty for almost all s ∈ [0, 1 − 1/p]. Moreover, he showed that the set Z 3/2 (s, s + 1/3) 2000 Mathematics Subject Classification: 11J71, 68R15.

Extremal properties of (epi)Sturmian sequences and distribution modulo $1$

“…While writing this survey we came across several extra references that are related to some of its parts. We give some of them here; the interested reader can look at these papers and the references therein: about combinatorics of words and Lorenz maps [10,11,12,13,55,79], about extremal properties of Sturmian sequences or measures [47,57,58], about the distribution of {ξα n } [2,3,31,85,86], and last but not least the historical paper of Lorenz [64] (also see [80]).…”

Squares and cubes in Sturmian sequences

“…We do not know whether D(S) = 2 or D(S) > 2. The inequality D(S) > 2 (if proved) has some applications to Mahler's problem: one can use the same method as in [14].…”

“…It is remarked in [1] that if α is an arbitrary algebraic number then for the same conclusion a somewhat stronger condition on the word ω is required. The paper [14] related to an unsolved Mahler's problem [21] about the powers of 3/2 modulo 1 is another example where this kind of information is necessary for Sturmian words ω. More precisely, in [14] one needs to estimate the smallest value of the supremum sup σ 0, τ 2 τ +σ 1+σ taken over all Sturmian words ω, where ω has infinitely many prefixes of the form uv τ , with |u| σ|v|.…”

Gap statistics and higher correlations for geometric progressions modulo one

“…A well known and open problem due to Mahler asks for the range of ({ξ(3/2) n }) n≥1 , where ξ > 0 is a real parameter. For more on this topic, and the study of the sequence (α n ) n≥1 modulo one, we refer the reader to [5,6,7,10] and the references therein.…”