An improvement of Konstantoulas’ density constant
Acta Arithmetica
MSC: Primary 11B34; Secondary 11B13
DOI: 10.4064/aa260528-23-7
Published online: 10 October 2026
Abstract
Let $A\subseteq \mathbb N$, and define its ordered representation function $$ r(n)=\#\{(a,b)\in A\times A:a+b=n\}. $$ The Erdős–Turán conjecture asserts that, if $r(n) \gt 0$ for all sufficiently large $n$, then $r(n)$ is unbounded. Konstantoulas proved a density-theoretic version: if the upper density of $E=\mathbb N \setminus (A+A)$ is less than $1/10$, then $\limsup _{n\to \infty } r(n) \gt 5$. In this paper, we improve Konstantoulas’ constant to $7/32$. We also prove that if $E$ has upper density less than $1$, then $\limsup _{n\to \infty } r(n) \gt 3$, and give a conditional criterion forcing $\limsup _{n\to \infty } r(n) \gt 7$.
