Note on unique representation bases

TL;DR

This paper improves the lower bound on the constant for unique representation bases of integers, showing that the minimal upper bound is at least 1, advancing understanding of the structure of such bases.

Contribution

The paper refines the lower bound for the constant in the unique representation basis problem, strengthening previous results and providing new insights into the bounds of these bases.

Findings

Improved lower bound for the constant c to at least 1

Enhanced understanding of the structure of unique representation bases

Progress in solving a problem posed by Chen in 2007

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 $c_{\mathscr{A}}\ge 1$.