Average sizes of mixed character sums

We prove that the average size of a mixed character sum \begin{equation*}\sum_{1\leqslant n \leqslant x} \chi(n) e(n\theta) w(n/x)\end{equation*} (for a suitable smooth function $w$ ) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine condition, where $\chi$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\leqslant r$ . In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $\theta$ . Certain quadratic Diophantine equations play a key role in the present paper.