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 $A(-x,x)=\big|A\cap[-x, x]\big|$. 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 $c_{\mathscr{A}}\ge 1$.
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Combinatorial Mathematics
Limits and Structures in Graph Theory
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 8 February 2026
- Language
- EN
- Document type
- Preprint