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 \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.