Bounding the exponential sum on squares of some sifted sequences

Abstract

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(\alpha; N)=\sum_{n\le N}b(n)e(n^2\alpha)$ satisfies \begin{equation*} \frac{S(\alpha;N)}{N/\sqrt{\log N}} \ll N^\epsilon (q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|\alpha - a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.