An Improvement of Konstantoulas' Density Constant

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 ${\mathrm{limsup}}_{n\to \infty }r(n)>5$. In this paper, we improve Konstantoulas' constant to $7/32$. We also prove that $D(E)<1/2$ implies ${\mathrm{limsup}}_{n\to \infty }r(n)>3$, and give a conditional criterion forcing ${\mathrm{limsup}}_{n\to \infty }r(n)>7$.

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