An Improvement of Konstantoulas' Density Constant

Let $A\subset N$, and define its ordered representation function $r(n)=\mathrm{\#}\{(a,b)\in A\times A:a+b=n\}\mathrm{.}$ 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 $1\mathrm{/}10$, then ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}r(n)>5$. In this paper, we improve Konstantoulas’ constant to $7\mathrm{/}32$. We also prove that $D(E)<1\mathrm{/}2$ implies ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}r(n)>3$, and give a conditional criterion forcing ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}r(n)>7$.