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 language | English |
|---|---|
| Pages (from-to) | 253-277 |
| Number of pages | 25 |
| Journal | Journal d'Analyse Mathematique |
| Volume | 133 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Oct 1 2017 |
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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 journal › Article
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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.
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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 -