ec 2 00 5 The Ternary Expansions of Powers of 2

P. Erdős asked how frequently the ternary expansion of 2 omits the digit 2. He conjectured this holds only for finitely many values of n. We generalize this question to iterates of two discrete dynamical systems. We consider real sequences xn(λ) = ⌊λ2 ⌋, where λ > 0 is a real number, and 3-adic integer sequences yn(λ) = λ2 , where λ is a 3-adic integer. We show that the set of initial values having infinitely many iterates that omit the digit 2 is small in a suitable sense, and 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.