Dense finite Sidon sets on arithmetic progressions

Abstract

Let $S\subset \{1,2,\dots ,n\}$ be a Sidon set with $|S|=n^{1/2}+O(n^{1/2-\hat{I}´})$ for some fixed $\hat{I}´>0$. This article provides the following expected asymptotic formula $$\sum _{\begin{array}{c}a\in S\\ a\equiv r(\mathrm{mod}m)\end{array}}{a}^{\ell }=\frac{1}{m(\ell +1)}{n}^{\ell +1/2}+o\left(n^{\ell +1/2}\right),$$ where $m\ge 1$, $0\le r<m$, and $\ell \ge 0$ are three integers. This removes the additional hypothesis in a previous residue class asymptotic formula by the author. The proof uses the Fourier uniformity of extremal Sidon sets due to Ortega and Prendiville.