Distinct distances on hyperbolic surfaces

Abstract

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 $\ge C_{\Gamma }\frac{N}{\log N}$ 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 $\ge C\frac{N}{\mu \log N}$ distinct distances for some absolute constant $C>0$.

References in corpus (1)

• Erdős distinct distances in hyperbolic surfaces