Powers in prime bases and a problem on central binomial coefficients

It is an open problem whether $\left(\genfrac{}{}{0px}{}{2n}{n}\right)$ 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 $m\nmid \left(\genfrac{}{}{0px}{}{2^{k+1}}{2^k}\right)$ 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 $⟨{\sigma }_t,\dots ,{\sigma }_1,{\sigma }_0⟩$, where $0\le {\sigma }_i<p$ for $i\ge 1$ 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.