Approximation orders of real numbers by $$\beta $$-expansions

Abstract

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their β-expansions with the exponential order β −n . Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under β-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of β-expansions.

A proof of the Conjecture of Lehmer

“…for almost all β > 1 (for the Lebesgue measure), was studied by Persson and Schmeling [148] [174], Ban and Li [16], Cao [49], Fang, Wu and Li [76], Li and Chen [118], Lü and Wu [126], Tan and Wang [196]. Kwon [108] studies the subset of Parry numbers whose conjugates lie close to the unit circle, using technics of combinatorics of words.…”

Distributions of full and non-full words in beta-expansions

“…Therefore {1, 2, · · · , k} ⊂ {τ β (s) : 1 ≤ s ≤ n}. For n ∈ N, we use r n (β) to denote the maximal length of the strings of 0's in ǫ * 1 · · · ǫ * n as in [FWL16], [HTY16] and [TYZ16], i.e., r n (β) = max{k ≥ 1 : ǫ * i+1 = · · · = ǫ * i+k = 0 for some 0 ≤ i ≤ n − k} with the convention that max ∅ = 0.…”

Exact Diophantine approximation of real numbers by $ \beta $-expansions

“…Recently, Fang et al [10] considered the approximation order of a real number by its convergents. More precisely, for a real number x ∈ [0, 1], we call the partial sums of the series (1)…”

“…Roughly speaking, almost all real numbers x can be approximated by their convergents ω n (x) with exponential order β −n . At the same time, they also considered the set of real numbers which can be approximated with orders β −φ(n) for a given positive function φ by showing that Theorem 1.2 (See [10]). If φ is non-decreasing and satisfies…”

“…In [10], the authors gave the approximation orders of real numbers by β-expansions, i.e. for λ-almost all x ∈ [0, 1)…”