Note on unique representation bases

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

💡 Research Summary

The paper investigates the growth rate of unique representation bases (URBs) for the integers. A set (A\subset\mathbb Z) is a URB if every integer (n) can be written uniquely as a sum (n=a+a’) with (a\le a’) and (a,a’\in A). For a URB (A) define \