SOLVED

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?

Lindström [Li98] has shown this is true for $A$ itself, subsequently strengthened by Kolountzakis [Ko99]. It follows immediately using the Sidon property that $A+A$ is similarly well-distributed.