# 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 $Z_{3}$ given by multiplication by $2$. The exceptional set $E(Z_{3})$ is defined to be the set of all elements of $Z_{3}$ whose forward orbits under this action intersect the $3$-adic Cantor set $Σ_{3,2ˉ}$ (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 $21 $, 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 $S=Σ_{3,2ˉ}∩M_{1}1 Σ_{3,2ˉ}∩⋯∩M_{n}1 Σ_{3,2ˉ}$ by integers $1<M_{1}<M_{2}<⋯<M_{n}$. 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 $(M_{1},…,M_{n})$ and to compute the Hausdorff dimension, which is always of the form $g_{3}(β)$ where $β$ 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 $M_{1},…,M_{n}$. 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