Counting non-standard binary representations

TL;DR

This paper investigates the asymptotic behavior of the number of representations of integers in non-standard binary systems, revealing that the summatory function grows exponentially with a rational coefficient.

Contribution

It provides a new asymptotic approximation for the summatory function of non-standard binary representations, including explicit growth rates and rational constants.

Findings

The summatory function $s_ ext{A}(r,m)$ grows approximately as $c( ext{A},m)| ext{A}|^r$.

The constant $c( ext{A},m)$ is rational.

The asymptotic behavior holds for a broad class of digit sets $ ext{A}$.

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 $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. We consider asymptotics of the summatory function $s_\mathcal{A}(r,m)$ of $f_\mathcal{A}(n)$ from $m2^r$ to $m2^{r+1}-1$ and show that $s_{\mathcal{A}}(r,m)\approx c(\mathcal{A},m)\left|\mathcal{A}\right|^r$ for some $c(\mathcal{A},m)\in\mathbb{Q}$.