Power and rank-weighted sums in dense finite Sidon sets

Abstract

Let $S\subset \{1,2,\ldots,n\}$ be a Sidon set with $|S|=n^{1/2}+O(n^{1/2-δ})$ for some fixed $δ>0$. This article provides the following expected asymptotic formula $$ \sum_{\substack{a\in S\\ a\equiv r\pmod{m}}} a^\ell =\frac{1}{m(\ell+1)}n^{\ell+1/2} +o\left(n^{\ell+1/2}\right), $$ where $m\geq 1$, $0\leq r<m$, and $\ell\geq 0$ 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.