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

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

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 languageAmerican English
Pages (from-to)253-277
Number of pages25
JournalJournal d'Analyse Mathematique
Volume133
Issue number1
DOIs
StatePublished - 1 Oct 2017

Bibliographical note

Publisher Copyright:
© 2017, Hebrew University Magnes Press.

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