论文标题
离散晶格轨道的对与veech表面的应用
Pairs in discrete lattice orbits with applications to Veech surfaces
论文作者
论文摘要
令$λ_1$，$λ_2$为晶格$γ<\ mathrm {sl} _2（\ Mathbb {r}）$的线性动作下的两个离散轨道。我们证明了Siegel $ - $ veech-type的积分公式，用于平均$$ \ sum _ {\ MathBf {\ MathBf {x} \inλ_1} \ sum _ {\ sum _ {\ Mathbf {y} \inλ_2} $ s_m $ $ $ $ M $ $ m $的鞍形连接的载体。这包括有关线性转换的通用鲍勒集合的有效计数，以及$ s_m $的对上的上限，带有有限的决定因素和$ s_m $的对$ s_m $的数量。该最后一个估计在附录中用于证明，几乎每$（θ，ψ）\在s^1 \ times s^1 $中，转换流量$f_θ^t $和$ f_in^t $在任何Veech Surface $ m $上都是不相同的。
Let $Λ_1$, $Λ_2$ be two discrete orbits under the linear action of a lattice $Γ<\mathrm{SL}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$ \sum_{\mathbf{x}\inΛ_1} \sum_{\mathbf{y}\inΛ_2} f(\mathbf{x}, \mathbf{y}) $$ from which we derive new results for the set $S_M$ of holonomy vectors of saddle connections of a Veech surface $M$. This includes an effective count for generic Borel sets with respect to linear transformations, and upper bounds on the number of pairs in $S_M$ with bounded determinant and on the number of pairs in $S_M$ with bounded distance. This last estimate is used in the appendix to prove that for almost every $(θ,ψ)\in S^1\times S^1$ the translations flows $F_θ^t$ and $F_ψ^t$ on any Veech surface $M$ are disjoint.
