Conjectured by Erd\H{o}s and Tur\'{a}n. They also suggest the stronger conjecture that $\limsup 1_A\ast 1_A(n)/\log n>0$.

Another stronger conjecture would be that the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.

Erd\H{o}s and S\'{a}rk\"{o}zy conjectured the stronger version that if $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ with $a_n/b_n\to 1$ are such that $A+B=\mathbb{N}$ then $\limsup 1_A\ast 1_B(n)=\infty$.

See also [40].

This is discussed in problem C9 of Guy's collection \cite{Gu04}.


References


[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.