JournalsjfgVol. 1, No. 2pp. 221–241

Distribution of full cylinders and the Diophantine properties of the orbits in β\beta-expansions

  • Yann Bugeaud

    Université de Strasbourg, France
  • Bao-Wei Wang

    Huazhong University of Science and Technology, Wuhan, China
Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions cover
Download PDF

Abstract

Let β>1\beta>1 be a real number. Let TβT_{\beta} denote the β\beta-transformation on [0,1)[0,1). A cylinder of order nn is a set of real numbers in [0,1)[0, 1) having the same first nn digits in their β\beta-expansion. A cylinder is called full if it has maximal length, i.e., if its length is equal to βn\beta^{-n}. In this paper, we show that full cylinders are well distributed in [0,1)[0,1) in a suitable sense. As an application to the metrical theory of β\beta-expansions, we determine the Hausdorff dimension of the set

{x[0,1]:Tβnxzn<eSnf(x)  for infinitely many  nN},\bigg\{x\in [0,1]: |T_{\beta}^nx-z_n|<e^{-S_nf(x)} \ {\text{ for infinitely many }}\ n\in \mathbb{N}\bigg\},

where {zn}n1\{z_n\}_{n\ge 1} is a sequence of real numbers in [0,1][0,1], the function f:[0,1]R+f:[0,1]\to \R^+ is continuous, and Snf(x)S_nf(x) denotes the ergodic sum f(x)++f(Tβn1x)f(x)+\cdots+f(T^{n-1}_{\beta}x).

Cite this article

Yann Bugeaud, Bao-Wei Wang, Distribution of full cylinders and the Diophantine properties of the orbits in β\beta-expansions. J. Fractal Geom. 1 (2014), no. 2, pp. 221–241

DOI 10.4171/JFG/6