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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S. K. was supported by EPSRC through grant EP/H000091/1.
D. K. was supported by the National Science Foundation through grant DMS-1101320. His stay at MSRI was supported in part by the NSF grant no. 0932078 000.
E. L. was supported by the European Research Council through AdG Grant 267259 and by the ISF grant 983/09. His stay at MSRI was supported in part by the NSF grant no. 0932078 000.
G. M. was supported by the National Science Foundation through grant DMS-1265695.
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Kadyrov, S., Kleinbock, D., Lindenstrauss, E. et al. Singular systems of linear forms and non-escape of mass in the space of lattices. JAMA 133, 253–277 (2017). https://doi.org/10.1007/s11854-017-0033-4
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DOI: https://doi.org/10.1007/s11854-017-0033-4