Erdős problem 154

Let $A\subset \{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?

The answer is yes. Lindström [Li98] proved the analogous statement for $A$ itself (see erdos_154.variants.lindstrom), later strengthened by Kolountzakis [Ko99]; well-distribution of $A+A$ then follows using the Sidon property.

We state the question for the sumset: for any sequence of Sidon sets $A_k\subseteq\{0,\ldots,N_k\}$ with $N_k\to\infty$ and $\lvert A_k\rvert\sim N_k^{1/2}$, and any modulus $m\geq 2$, the proportion of elements of $A_k+A_k$ congruent to $i\pmod m$ (i.e. the count divided by $\lvert A_k+A_k\rvert$) tends to $1/m$ for every residue $i<m$.