Erdős Problem #154 — Distribution of Sidon sumsets in residue classes

Let $m\geq 2$, let $N_k\to\infty$ be a sequence of natural numbers, and let $A_k\subseteq\{0,\ldots,N_k\}$ be Sidon sets, meaning that equal sums of two elements determine the same unordered pair. If

$$\frac{|A_k|}{\sqrt{N_k}}\longrightarrow 1,$$

then, for every residue $i<m$,

$$\frac{\bigl|\{s\in A_k+A_k:s\equiv i\pmod m\}\bigr|}{|A_k+A_k|}\longrightarrow\frac1m.$$