Intersections of multiplicative translates of 3-adic Cantor sets

Abstract

This paper is motivated by questions concerning the discrete dynamical system on the $3$-adic integers ${\mathbb{Z}}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}({\mathbb{Z}}_3)$ is defined to be the set of all elements of ${\mathbb{Z}}_3$ whose forward orbits under this action intersect the $3$-adic Cantor set ${\Sigma }_{3,\bar{2}}$ (of $3$-adic integers whose expansions omit the digit $2$) infinitely many times. It has been shown that this set has Hausdorff dimension at most $\frac{1}{2}$, and it is conjectured that it has Hausdorff dimension $0$. Upper bounds on its Hausdorff dimension can be obtained with sufficient knowledge of Hausdorff dimensions of intersections of multiplicative translates of Cantor sets by powers of $2$. This paper studies more generally the structure of finite intersections of general multiplicative translates $S={\Sigma }_{3,\bar{2}}\cap \frac{1}{M_1}{\Sigma }_{3,\bar{2}}\cap \cdots \cap \frac{1}{M_n}{\Sigma }_{3,\bar{2}}$ by integers $1<M_1<M_2<\cdots <M_n$. These sets are describable as sets of $3$-adic integers whose $3$-adic expansions have one-sided symbolic dynamics given by a finite automaton. This paper gives a method to determine the automaton for given data $(M_1,\dots ,M_n)$ and to compute the Hausdorff dimension, which is always of the form ${\log }_3(\beta )$ where $\beta$ is an algebraic integer. Computational examples indicate that in general the Hausdorff dimension of such sets depends in a very complicated way on the integers $M_1,\dots ,M_n$. Exact answers are obtained for certain infinite families, which show as a corollary that a relaxed notion of generalized exceptional set has a positive Hausdorff dimension.