For what functions $g(N) → \infty$ is it true that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

If we don't pose additional conditions on the functions, then this is a stronger form of the Erdős-Turán conjecture, see Erdõs Problem 28, (since establishing this for any function $g(N) → \infty$ would imply a positive solution to Erdős Problem 28).