Ternary expansions of powers of 2

Abstract

Erdős asked how frequently 2n has a ternary expansion that omits the digit 2. He conjectured that this holds only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first considers truncated ternary expansions of real sequences xn (λ) = ⌊λ2n ⌋, where λ > 0 is a real number, along with its untruncated version, whereas the second considers 3-adic expansions of sequences yn(λ) = λ2n, where λ is a 3-adic integer. We show in both cases that the set of initial values having infinitely many iterates that omit the digit 2 is small in a suitable sense. For each nonzero initial value we obtain an asymptotic upper bound as k → ∞ on the number of the first k iterates that omit the digit 2. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3-adic Cantor sets.