题目：Distinct distances on hyperbolic surfaces

主讲人：孟宪昌（University of Göttingen）

摘要：Erdos in 1946 asked the question of finding the minimal number of distinct distances among any $N$ points in the plane. In our work, we give complete answer for this problem on all hyperbolic surfaces of finite volume. For any cofinite Fuchsian group $\Gamma\subset\mathrm{PSL}(2, \mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $\Gamma\backslash\mathbb{H}^2$ determines $\geq C_{\Gamma} \frac{N}{\log N}$ distinct distances for some constant $C_{\Gamma}>0$ depending only on $\Gamma$. In particular, for $\Gamma$ being any finite index subgroup of $\mathrm{PSL}(2, \mathbb{Z})$ with $\mu=[\mathrm{PSL}(2, \mathbb{Z}): \Gamma ]<\infty$, any set of $N$ points on $\Gamma\backslash\mathbb{H}^2$ determines $\geq C\frac{N}{\mu\log N}$ distinct distances for some absolute constant $C>0$.

时间：2020年12月01日，14:00-15:00

地点：腾讯会议，会议 ID：995 338 269

邀请人：黄炳荣