Note on unique representation bases

Abstract

Answering affirmatively a 2007 problem of Chen, the first author proved that there is a unique representation basis $A$ of $\mathbb{Z}$ and a constant $c>0$ such that $$A(-x,x)\ge c\sqrt{x}$$ for infinitely many positive integers $x$, where $A(-x,x)=|A\cap [-x,x]|$. Let $c_{\mathcal{A}}$ be the least upper bound for such $c$. It was proved in the former article by the first author that $\sqrt{2}/2\le c_{\mathcal{A}}\le \sqrt{2}$. In this note, the prior result is improved to $c_{\mathcal{A}}\ge 1$.