Dense finite Sidon sets on arithmetic progressions

Let $S\subset \{1,2,\dots ,n\}$ be a Sidon set with $\mathrm{\mid }S\mathrm{\mid }=n^{1\mathrm{/}2}+O(n^{1\mathrm{/}2-\delta })$ for some fixed $\delta >0$. This article provides the following expected asymptotic formula $$\sum _{\begin{array}{c}a\in S\\ a\equiv r(\mathrm{m}\mathrm{o}\mathrm{d}m)\end{array}}{a}^{\mathrm{\ell }}=\frac{1}{m(\mathrm{\ell }+1)}{n}^{\mathrm{\ell }+1\mathrm{/}2}+o\left(n^{\mathrm{\ell }+1\mathrm{/}2}\right),$$ where $m\ge 1$, $0\le r<m$, and $\mathrm{\ell }\ge 0$ 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.