Note on unique representation bases

TL;DR

This work addresses the maximal growth of the centered representation count for unique representation bases of the integers. It proves a stronger lower bound $c_{\mathscr{A}}\ge 1$ by constructing a unique representation basis via an inductive process that interleaves Sidon-set insertions with careful control of representations, leveraging Bose-typeSidon-set results and modular-sum distinctness arguments. The approach tightens the connection between Sidon-set density and the growth of $A(-x,x)$, improving prior bounds $\sqrt{2}/2\le c_{\mathscr{A}}\le \sqrt{2}$ and suggesting the conjecture $c_{\mathscr{A}}=\sqrt{2}$. The results illuminate the density-structure tradeoffs for additive bases in $\mathbb{Z}$ and advance understanding of how sparse unique representation bases can be while maintaining full representability.

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

Theorems & Definitions (6)

• Theorem 1
• Proposition 1
• Proposition 2
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:1']}