Powers in prime bases and a problem on central binomial coefficients

It is an open problem whether $\binom{2n}{n}$ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $m \nmid \binom{2^{k+1}}{2^{k}}$ is 0. To do so we examine numbers of the form $\alpha^{k}$ in base $p$, where $p$ is a prime and $(\alpha, p)=1$. For every $n$ and $a$ we find an upper bound on the number of $k$'s less than $a$ such that $(\alpha^{k})_p$ contains less than $n$ digits greater than $\frac{p}{2}$. This is done by showing that every sequence of the form $\langle \sigma_t, \dots, \sigma_1,\sigma_0 \rangle$, where $0\leq \sigma_i<p$ for $i\geq 1$ and $\sigma_0$ is in the residue class generated by $\alpha$ modulo $p$, occurs at specific places in the representation $(\alpha^k)_p$ as $k$ varies.