Diophantine approximation on matrices and Lie groups

被引:13
作者
Aka, Menny [1 ]
Breuillard, Emmanuel [2 ,3 ]
Rosenzweig, Lior [4 ]
de Saxce, Nicolas [5 ]
机构
[1] ETH, Dept Math, Ramistr 101, CH-8092 Zurich, Switzerland
[2] Univ Munster, Math Inst, 62 Einsteinstr, Munster, Germany
[3] DPMMS, Wilberforce Rd, Cambridge CB3 0WB, England
[4] ORT Braude Coll, Dept Math, POB 78, IL-21982 Karmiel, Israel
[5] Univ Paris 13, CNRS, LAGA, F-93430 Villetaneuse, France
基金
欧洲研究理事会;
关键词
Metric diophantine approximation; Homogeneous dynamics; Extremal manifolds; Group actions; HOMOGENEOUS SPACES; THEOREM; FLOWS; DYNAMICS; SYSTEMS; GROWTH;
D O I
10.1007/s00039-018-0436-0
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study the general problem of extremality for metric diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over , we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest. In particular we prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.
引用
收藏
页码:1 / 57
页数:57
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