Large values of shifted mixed character sums

Abstract

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