2 problems

Let $B\in\mathbb{N}_{\ge 2}$ and let $D\subseteq\{0,1,\dots,B-1\}$ satisfy $|D|\ge 2$. Let $\alpha_B^D(n)$ be the number of integers $B^{n-\frac12}\le x<B^{n+\frac12}$ such that th…

For a base $B\ge 2$ and a digit set $D\subseteq\{0,1,\dots,B-1\}$ with $|D|\ge 2$, let $\omega_B^D(n)$ denote the number of $n$-digit suffixes in base $B$ from $D$ that occur as th…