Large values of shifted mixed character sums

Abstract

We consider sums of the form $$F_χ(α,β;θ) := \sum_{αp<n\leβp}χ(n)e(nθ),$$ where $χ$ is a non-principal Dirichlet character modulo a prime number $p$. We prove that $$ \sqrt p \log \log p \ll \max_{0 \le θ< 1}{\left|F_χ(α,β;θ)\right|} \ll \sqrt{p}\log p, $$ generalizing an old result of Montgomery as well as a recent result of Iggidr in two aspects: we allow general non-principal characters $χ$, and we consider incomplete mixed character sums.