Unique expansions and intersections of Cantor sets

To each $\alpha \in (1\mathrm{/}3,1\mathrm{/}2)$ we associate the Cantor set $${\mathrm{\Gamma }}_{\alpha }:=\{\sum _{i=1}^{\mathrm{\infty }}{ϵ}_i{\alpha }^i:{ϵ}_i\in \{0,1\},i\ge 1\}\mathrm{.}$$ In this paper we consider the intersection $\Gamma_\alpha \cap (\Gamma_\alpha

• t)$foranytranslation$t\in$\mathbb{R}$$\mathrm{.}Wepayspecialattentiontothose$t$withaunique${-1,0,1}$ $\alpha$-expansion, and study the set $$D_{\alpha }:=\{{\dim }_H({\mathrm{\Gamma }}_{\alpha }\cap ({\mathrm{\Gamma }}_{\alpha }+t)):t\text{ has a unique }\{-1,0,1\}\alpha \text{-expansion}\}\mathrm{.}$$ We prove that there exists a transcendental number ${\alpha }_{KL}\approx 0.39433\dots$ such that: $D_{\alpha }$ is finite for $\alpha \in ({\alpha }_{KL},1\mathrm{/}2),$ $D_{{\alpha }_{KL}}$ is infinitely countable, and $D_{\alpha }$ contains an interval for $\alpha \in (1\mathrm{/}3,{\alpha }_{KL})\mathrm{.}$ 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 ${\dim }_H({\mathrm{\Gamma }}_{\alpha }\cap ({\mathrm{\Gamma }}_{\alpha }+t))$ when ${\mathrm{\Gamma }}_{\alpha }\cap ({\mathrm{\Gamma }}_{\alpha }+t)$ 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 ${\dim }_H({\mathrm{\Gamma }}_{\alpha }\cap ({\mathrm{\Gamma }}_{\alpha }+t)),$ and for which ${\mathrm{\Gamma }}_{\alpha }\cap ({\mathrm{\Gamma }}_{\alpha }+t)$ is a self-similar set. We also construct $\alpha$ and $t$ for which ${\mathrm{\Gamma }}_{\alpha }\cap ({\mathrm{\Gamma }}_{\alpha }+t)$ 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.