
Mathematics 2013
Intersections of multiplicative translates of 3adic Cantor setsAbstract: Motivated by a question of Erd\H{o}s, this paper considers questions concerning the discrete dynamical system on the 3adic integers given by multiplication by 2. Let the 3adic Cantor set consist of all 3adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3adic integers whose forward orbits under this action intersects the 3adic 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 3adic Cantor set by integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3adic integers whose 3adic expansions have onesided 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.
