Univoque bases of real numbers: simply normal bases, irregular bases and multiple rationals

TL;DR

This paper investigates the properties of univoque bases for real numbers, showing that sets of bases with unique simply normal or irregular expansions have full Hausdorff dimension, and establishing the existence of bases with unique expansions for finitely many rationals.

Contribution

It proves that the sets of univoque simply normal and irregular bases for any real number have full Hausdorff dimension, and constructs bases with unique expansions for multiple rationals.

Findings

Sets of univoque simply normal bases have full Hausdorff dimension.

Sets of univoque irregular bases have full Hausdorff dimension.

Existence of bases with unique expansions for finitely many rationals.

Abstract

Given a positive integer $M$ and a real number $x\in(0,1]$, we call $q\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\in\{0,1,\ldots,M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$. Similarly, a base $q\in(1,M+1]$ is called a univoque irregular base of $x$ if there exists a unique sequence $(d_i)\in\{0,1,\ldots, M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$ and the sequence $(d_i)$ has no digit frequency. Let $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ be the sets of univoque simply normal bases and univoque irregular bases of $x$, respectively. In this paper we show that for any $x\in(0,1]$ both $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ have full Hausdorff dimension. Furthermore, given finitely many rationals $x_1, x_2, \ldots, x_n\in(0,1]$ so that each $x_i$ has a finite expansion in base $M+1$, we show that there exists a full Hausdorff dimensional set of $q\in(1,M+1]$ such that each $x_i$ has a unique expansion in base $q$.