Ternary expansions of powers of 2

Abstract

Paul Erdos asked how frequently the ternary expansion of 2^n omits the digit 2. He conjectured this happens only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first is over the real numbers, and considers the integer part of lambda 2^n for a real input lambda. The second is over the 3-adic integers, and considers the sequence lambda 2^n for a 3-adic integer input lambda. We show that the number of input values that have infinitely many iterates omitting the digit 2 in their ternary expansion is small in a suitable sense. For each nonzero input we give an asymptotic upper bound on the number of the first k iterates that omit the digit 2, as k goes to infinity. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3-adic Cantor sets.