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Limiting distributions of translates of divergent diagonal orbits

Part of: Ergodic theory Multiplicative number theory Noncompact transformation groups

Published online by Cambridge University Press:  02 August 2019

Uri Shapira  and
Cheng Zheng
Uri Shapira
Affiliation:
Department of Mathematics, Technion, Haifa, Israel email [email protected]
Cheng Zheng
Affiliation:
Department of Mathematics, Technion, Haifa, Israel email [email protected]
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Abstract

We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite-volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity.

Keywords

homothety classes of locally finite measuresconvex polytopestranslates of divergent orbitshomogeneous spacesRatner’s theoremcounting lattice points

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 22F30: Homogeneous spaces 11N45: Asymptotic results on counting functions for algebraic and topological structures
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 9 , September 2019 , pp. 1747 - 1793
Copyright
© The Authors 2019 

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Footnotes

The authors acknowledge the support of ISF grants 871/17, 662/15 and 357/13. The second author is in part supported at the Technion by a Fine Fellowship. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 754475).

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