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arXiv · 2602.07743

Note on unique representation bases

Yuchen Ding·Jie Wang
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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 $A(-x,x)=\big|A\cap[-x, x]\big|$. Let $c_{\mathscr{A}}$ be the least upper bound for such $c$. It was proved in the former article by the first author that $\sqrt{2}/2\le c_{\mathscr{A}}\le \sqrt{2}$. In this note, the prior result is improved to $c_{\mathscr{A}}\ge 1$.

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Yuchen Ding, Jie Wang. 2026-02-08. Note on unique representation bases. https://arxiv.org/abs/2602.07743

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