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 $$\sqrt{p}\log \log p\ll \underset{0\le \theta <1}{\max }\left|F_{\chi }(\alpha ,\beta ;\theta )\right|\ll \sqrt{p}\log p,$$ 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.