# Intersections of multiplicative translates of 3-adic Cantor sets

### William C. Abram

Hillsdale College, USA### Jeffrey C. Lagarias

University of Michigan, Ann Arbor, USA

## 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 $S= \Sigma_{3, \bar{2}} \cap \frac{1}{M_1} \Sigma_{3, \bar{2}} \cap \dots \cap \frac{1}{M_n} \Sigma_{3, \bar{2}}$ by integers $1 < M_1 < M_2 < \dots < M_n$. 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.

## Cite this article

William C. Abram, Jeffrey C. Lagarias, Intersections of multiplicative translates of 3-adic Cantor sets. J. Fractal Geom. 1 (2014), no. 4, pp. 349–390

DOI 10.4171/JFG/11