Counting non-standard binary representations

Abstract

Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_{\mathcal{A}}(n)$ be the number of ways to write $n={\sum }_{i=0}^{\infty }{\epsilon }_i{2}^i$, where ${\epsilon }_i\in \mathcal{A}$. We consider asymptotics of the summatory function $s_{\mathcal{A}}(r,m)$ of $f_{\mathcal{A}}(n)$ from $m{2}^r$ to $m{2}^{r+1}-1$ and show that $s_{\mathcal{A}}(r,m)\approx c(\mathcal{A},m){\left|\mathcal{A}\right|}^r$ for some $c(\mathcal{A},m)\in \mathbb{Q}$.