Unique expansions and intersections of Cantor sets

Abstract

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