论文标题
双曲线表面上的不同距离
Distinct distances on hyperbolic surfaces
论文作者
论文摘要
对于任何CofInite fuchsian $γ\ subset {\ rm psl}（2，2，\ mathbb {r}）$，我们表明，双曲表面$γ\ backslash \ backslash \ backslash \ mathbb \ mathbb {h}^h}^2 $ for y Mathbb {h}^$ geq c_ geqc_γ\ frac \ frac \ frac {n}的任何集合$ n $点$C_γ> 0 $仅取决于$γ$。特别是，对于$γ$，是$ {\ rm psl}的任何有限索引子组（2，2，\ mathbb {z}）$，$μ= [{\ rm psl}（\ rm psl}（2，\ mathbb {z}） $ \ geq c \ frac {n} {μ\ log n} $独特的距离，对于某些绝对常数$ c> 0 $。
For any cofinite Fuchsian group $Γ\subset {\rm PSL}(2, \mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $Γ\backslash\mathbb{H}^2$ determines $\geq C_Γ \frac{N}{\log N}$ distinct distances for some constant $C_Γ>0$ depending only on $Γ$. In particular, for $Γ$ being any finite index subgroup of ${\rm PSL}(2, \mathbb{Z})$ with $μ=[{\rm PSL}(2, \mathbb{Z}): Γ]<\infty$, any set of $N$ points on $Γ\backslash\mathbb{H}^2$ determines $\geq C\frac{N}{μ\log N}$ distinct distances for some absolute constant $C>0$.
