An Improvement of Konstantoulas' Density Constant

Huixi Li; Zihan Zhang

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arXiv · 2605.30922

An Improvement of Konstantoulas' Density Constant

Huixi Li·Zihan Zhang

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Abstract

Let $A\subset N$, and define its ordered representation function $r(n)=\#\{(a,b)\in A\times A:a+b=n\}.$ The Erdos--Turan conjecture asserts that, if $r(n) > 0$ for all sufficiently large $n$, then $r(n)$ is unbounded. Konstantoulas proved a density-theoretic version: if the upper density of $E=N\setminus(A+A)$ is less than $1/10$, then $\limsup_{n\to\infty} r(n)> 5$. In this paper, we improve Konstantoulas' constant to $7/32$. We also prove that $D(E)< 1/2$ implies $\limsup_{n\to\infty} r(n) > 3$, and give a conditional criterion forcing $\limsup_{n\to\infty} r(n)>7$.

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Huixi Li, Zihan Zhang. 2026-05-29. An Improvement of Konstantoulas' Density Constant. https://arxiv.org/abs/2605.30922

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