Ergodic Theory on Homogeneous Spaces and Metric Number Theory
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
复制标题
齐次空间的遍历理论和度量数论
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
D. Kleinbock
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文献类型：
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作者：
D. Kleinbock
文章概述本文简要介绍了度量数论的最新发展，特别是流形上的丢番图逼近，它是通过应用齐次空间上动力学的思想和方法而获得的。词汇表1.定义：度量丢番图近似2.基本事实3。导言4.与动力学的连接上的空间格5.丢番图近似与相关量6.更多结果7丢番图逼近（Diophantine approximation）丢番图逼近（Diophantine approximation）是指用有理数来逼近真实的数，或者更一般地说，找到一些（可能是向量值的）函数达到接近整数的值的整数点。度量数论度量数论（或者，具体地说，度量丢番图逼近）是指研究具有规定的丢番图逼近性质的真实的数或向量的集合。齐性空间一个群G通过它的子群Γ的齐性空间G/Γ是陪集空间{gΓ}。当G是一个李群，而Γ是一个离散子群时，空间G/Γ是一个光滑流形，局部看起来像G本身。格;幺模格李群中的格是有限余体积的离散子群;幺模表示余体积等于1。
Article outline This article gives a brief overview of recent developments in metric number theory, in particular, Diophantine approximation on manifolds, obtained by applying ideas and methods coming from dynamics on homogeneous spaces. Glossary 1. Definition: Metric Diophantine approximation 2. Basic facts 3. Introduction 4. Connection with dynamics on the space of lattices 5. Diophantine approximation with dependent quantities 6. Further results 7. Future directions References Glossary Diophantine approximation Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector-valued) functions attain values close to integers. metric number theory Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties. homogeneous spaces A homogeneous space G/Γ of a group G by its subgroup Γ is the space of cosets {gΓ}. When G is a Lie group and Γ is a discrete subgroup, the space G/Γ is a smooth manifold and locally looks like G itself. lattice; unimodular lattice A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1.