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.Comment: v1, 31 pages, 6 figure