Multiple expansions of real numbers with digits set $\{0,1,q\}$

Abstract

For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let ${\mathcal{U}}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2,3,\cdots ,{\aleph }_0$ let ${\mathcal{B}}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in {\mathcal{B}}_k$ let ${\mathcal{U}}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \[ \mathcal{B}_{\aleph_0}=[2,\infty),\quad \mathcal{B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2, \] where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in {\mathcal{B}}_k$ the Hausdorff dimensions of ${\mathcal{U}}_q^{(k)}$ and ${\mathcal{U}}_q$ are the same, i.e., \[ \dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2. \] Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.

15 page, to appear in Mathematische Zeitschrift

References in corpus (3)

• Expansions in non-integer bases: lower, middle and top orders
• On a problem of countable expansions
• Numbers with countable expansions in base of generalized golden ratios

Cited by in corpus (4)

• Projections of cartesian products of the self-similar sets without the irrationality assumption
• Subshift of finite type and self-similar sets
• Multiplication on self-similar sets with overlaps
• Estimating the Hausdorff dimensions of univoque sets for self-similar sets