On character sums over flat numbers

Let $q⩾2$ be an integer, $\chi$ be any non-principal character mod $q$, and $H=H(q)⩽q\mathrm{.}$ In this paper the authors prove some estimates for character sums of the form $$\mathcal{W}(\chi ,H;q)=\sum _{n\in \mathcal{F}(H)}\chi (n),$$ where \mathscr{F}(H)=\left\{n\in\mathbb{Z}|(n,q)=1,1\leqslant n,\bar{n}\leqslant q, |n-\bar{n}|\leqslant H\}, $\bar{n}$ is defined by $n\bar{n}\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}q)\mathrm{.}$