Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 253–277 | Cite as

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

  • S. Kadyrov
  • D. Kleinbock
  • E. Lindenstrauss
  • G. A. Margulis
Article

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.

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Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  • S. Kadyrov
    • 1
  • D. Kleinbock
    • 2
  • E. Lindenstrauss
    • 3
  • G. A. Margulis
    • 4
  1. 1.Mathematics DepartmentNazarbayev UniversityAstanaKazakhstan
  2. 2.Department of MathematicsBrandeis UniversityWalthamUSA
  3. 3.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  4. 4.Department of MathematicsYale UniversityNew HavenUSA

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