Pairs in discrete lattice orbits with applications to Veech surfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913612188090368 |
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| author | Burrin, Claire Fairchild, Samantha Chaika, Jon |
| author_facet | Burrin, Claire Fairchild, Samantha Chaika, Jon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_14621 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pairs in discrete lattice orbits with applications to Veech surfaces Burrin, Claire Fairchild, Samantha Chaika, Jon Dynamical Systems Geometric Topology Number Theory 22E40, 37E35, 11F72 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. |
| title | Pairs in discrete lattice orbits with applications to Veech surfaces |
| topic | Dynamical Systems Geometric Topology Number Theory 22E40, 37E35, 11F72 |
| url | https://arxiv.org/abs/2211.14621 |