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 \(i<m\).