Limiting distributions of translates of divergent diagonal orbits

Uri Shapira; Cheng Zheng

doi:10.1112/S0010437X19007450

Limiting distributions of translates of divergent diagonal orbits

Uri Shapira, Cheng Zheng

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1747-1793
Number of pages	47
Journal	Compositio Mathematica
Volume	155
Issue number	9
DOIs	https://doi.org/10.1112/S0010437X19007450
State	Published - 1 Sep 2019

Keywords

Ratner's theorem
convex polytopes
counting lattice points
homogeneous spaces
homothety classes of locally finite measures
translates of divergent orbits

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1112/S0010437X19007450

Cite this

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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 = "Ratner's theorem, convex polytopes, counting lattice points, homogeneous spaces, homothety classes of locally finite measures, translates of divergent orbits",

author = "Uri Shapira and Cheng Zheng",

note = "Funding Information: 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{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement no. 754475). This journal is {\textcopyright}c Foundation Compositio Mathematica 2019.",

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N2 - 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.

AB - 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.

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KW - counting lattice points

KW - homogeneous spaces

KW - homothety classes of locally finite measures

KW - translates of divergent orbits

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JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 9

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Limiting distributions of translates of divergent diagonal orbits

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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