Unique expansions and intersections of Cantor sets

Abstract

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

19 pages, two figures