Singular systems of linear forms and non-escape of mass in the space of lattices

S. Kadyrov, D. Kleinbock, E. Lindenstrauss, G. A. Margulis

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.

Original languageEnglish
Pages (from-to)253-277
Number of pages25
JournalJournal d'Analyse Mathematique
Volume133
Issue number1
DOIs
Publication statusPublished - Oct 1 2017

Fingerprint

Singular Systems
Linear Forms
Trajectory
Integral Inequality
One to one correspondence
Hausdorff Dimension
Divergence
Orbit
Upper bound

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

Cite this

Singular systems of linear forms and non-escape of mass in the space of lattices. / Kadyrov, S.; Kleinbock, D.; Lindenstrauss, E.; Margulis, G. A.

In: Journal d'Analyse Mathematique, Vol. 133, No. 1, 01.10.2017, p. 253-277.

Research output: Contribution to journalArticle

Kadyrov, S. ; Kleinbock, D. ; Lindenstrauss, E. ; Margulis, G. A. / Singular systems of linear forms and non-escape of mass in the space of lattices. In: Journal d'Analyse Mathematique. 2017 ; Vol. 133, No. 1. pp. 253-277.
@article{aabe86e26ca64159b7270cc926116a03,
title = "Singular systems of linear forms and non-escape of mass in the space of lattices",
abstract = "Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.",
author = "S. Kadyrov and D. Kleinbock and E. Lindenstrauss and Margulis, {G. A.}",
year = "2017",
month = "10",
day = "1",
doi = "10.1007/s11854-017-0033-4",
language = "English",
volume = "133",
pages = "253--277",
journal = "Journal d'Analyse Mathematique",
issn = "0021-7670",
publisher = "Springer New York",
number = "1",

}

TY - JOUR

T1 - Singular systems of linear forms and non-escape of mass in the space of lattices

AU - Kadyrov, S.

AU - Kleinbock, D.

AU - Lindenstrauss, E.

AU - Margulis, G. A.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.

AB - Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of one-parameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of the set of singular systems of linear forms (equivalently, the set of lattices with divergent trajectories) as well as the dimension of the set of lattices with trajectories “escaping on average” (a notion weaker than divergence). This extends work by Cheung, as well as by Chevallier and Cheung. Our method differs considerably from that of Cheung and Chevallier and is based on the technique of integral inequalities developed by Eskin, Margulis and Mozes.

UR - http://www.scopus.com/inward/record.url?scp=85037054321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85037054321&partnerID=8YFLogxK

U2 - 10.1007/s11854-017-0033-4

DO - 10.1007/s11854-017-0033-4

M3 - Article

VL - 133

SP - 253

EP - 277

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -