Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres

IF 0.8 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems Pub Date : 2024-04-26 DOI:10.1017/etds.2024.31

WOOYEON KIM, JENS MARKLOF

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引用次数: 0

Abstract

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded

$\operatorname {SL}(d,\mathbb {R})$

-horospheres in the space of affine lattices.

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多维仿射网格中方向的泊松对相关性和嵌入角球的质量逃逸估计值

我们证明了在移位的自然戴奥芬汀条件下，仿射网格向量落入小圆盘的方向数矩的收敛性。此外，我们还证明，对于维度 3 及更高的任何无理平移，包括可近似的向量，对相关函数都是泊松的。分布的收敛性在斯特罗姆伯格森和第二作者的著作[周期洛伦兹气体中自由路径长度的分布及相关晶格点问题。Ann. of Math. (2)172 (2010), 1949-2033]，而扩展到时刻收敛的主要步骤是对仿射网格空间中嵌入 $\operatorname {SL}(d,\mathbb {R})$ -horospheres 的平均值进行质量逃逸估计。

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来源期刊

Ergodic Theory and Dynamical Systems 数学-数学

CiteScore

1.70

自引率

11.10%

发文量

113

审稿时长

6-12 weeks

期刊介绍： Ergodic Theory and Dynamical Systems focuses on a rich variety of research areas which, although diverse, employ as common themes global dynamical methods. The journal provides a focus for this important and flourishing area of mathematics and brings together many major contributions in the field. The journal acts as a forum for central problems of dynamical systems and of interactions of dynamical systems with areas such as differential geometry, number theory, operator algebras, celestial and statistical mechanics, and biology.