Unique expansions and intersections of Cantor sets

TL;DR

This paper investigates the Hausdorff dimension of intersections of Cantor sets generated by powers of a parameter alpha, focusing on unique expansions, and reveals complex behaviors depending on alpha's value, including finiteness, countability, and interval properties.

Contribution

It introduces a detailed analysis of the intersection dimensions of Cantor sets with unique alpha-expansions, identifying a transcendental threshold and characterizing when the dimension set forms an interval.

Findings

Existence of a transcendental alpha where the dimension set is countably infinite.

For alpha in a specific interval, the dimension set contains an entire interval.

The dimension set is finite or equals a specific interval depending on alpha's value.

Abstract

To each $\alpha\in(1/3,1/2)$ we associate the Cantor set $$\Gamma_{\alpha}:=\Big\{\sum_{i=1}^{\infty}\epsilon_{i}\alpha^i: \epsilon_i\in\{0,1\},\,i\geq 1\Big\}.$$ In this paper we consider the intersection $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ for any translation $t\in\mathbb{R}$. We pay special attention to those $t$ with a unique $\{-1,0,1\}$ $\alpha$-expansion, and study the set $$D_\alpha:=\{\dim_H(\Gamma_\alpha \cap (\Gamma_\alpha + t)):t \textrm{ has a unique }\{-1,0,1\}\,\alpha\textrm{-expansion}\}.$$ We prove that there exists a transcendental number $\alpha_{KL}\approx 0.39433\ldots$ such that: $D_\alpha$ is finite for $\alpha\in(\alpha_{KL},1/2),$ $D_{\alpha_{KL}}$ is infinitely countable, and $D_{\alpha}$ contains an interval for $\alpha\in(1/3,\alpha_{KL}).$ We also prove that $D_\alpha$ equals $[0,\frac{\log 2}{-\log \alpha}]$ if and only if $\alpha\in (1/3,\frac {3-\sqrt{5}}{2}].$ As a consequence of our investigation we prove some results on the possible values of $\dim_{H}(\Gamma_\alpha \cap (\Gamma_\alpha + t))$ when $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ is a self-similar set. We also give examples of $t$ with a continuum of $\{-1,0,1\}$ $\alpha$-expansions for which we can explicitly calculate $\dim_{H}(\Gamma_\alpha\cap(\Gamma_\alpha+t)),$ and for which $\Gamma_\alpha\cap (\Gamma_\alpha+t)$ is a self-similar set. We also construct $\alpha$ and $t$ for which $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ contains only transcendental numbers. Our approach makes use of digit frequency arguments and a lexicographic characterisation of those $t$ with a unique $\{-1,0,1\}$ $\alpha$-expansion.