Ju l 2 00 8 Ternary Expansions of Powers of 2 Jeffrey

P. Erdős asked how frequently does 2n have 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, while the second considers 3-adic expansions of sequences yn(λ) = λ2 n, 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 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.