Intersections of multiplicative translates of 3-adic Cantor sets

TL;DR

This paper investigates the structure and Hausdorff dimensions of intersections of multiplicative translates of the 3-adic Cantor set, revealing complex dependencies on the involved integers and providing a method to analyze their automata representations.

Contribution

It introduces a method to determine automata for intersections of multiplicative translates of the 3-adic Cantor set and explores the algebraic nature of their Hausdorff dimensions.

Findings

Hausdorff dimension is always log(eta) for algebraic eta

Dimensions depend in a complex way on the integers M_1,...,M_n

Provides a method to construct automata for these sets

Abstract

Motivated by a question of Erd\H{o}s, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Cantor sets by powers of 2. More generally, this paper studies the structure of finite intersections of general multiplicative translates of the 3-adic Cantor set 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. As a consequence, the Hausdorff dimension of such a set is always of the form log(\beta) for an algebraic integer \beta. This paper gives a method to determine the automaton for given data (M_1, ..., M_n). Experimental results indicate that the Hausdorff dimension of such sets depends in a very complicated way on the integers M_1,...,M_n.