Expansions of Powers of 2 Jeffrey

P. Erdős asked how frequently 2 has a ternary expansion that omits the digit 2. He conjectured this holds only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first considers ternary expansions of real sequences xn(λ) = ⌊λ2 ⌋, where λ > 0 is a real number, and the second considers 3-adic expansions of sequences yn(λ) = λ2 , 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.