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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References
J. Athreya. Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata 119 (2006), 121–140.
M. Brin and A. Katok, On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981), Springer, Berlin, 1983, pp. 30–38.
Y. Cheung, Hausdorff dimension of the set of singular pairs, Ann. of Math. (2), 173 (2011), 127–167.
Y. Cheung and N. Chevallier, Hausdorff dimension of singular vectors, Duke Math. J., 165 (2016), 2273–2329.
S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55–89.
M. Einsiedler and S. Kadyrov, Entropy and escape of mass for SL(3, Z)\SL(3,R), Israel J. Math., 190 (2012), 253–288.
M. Einsiedler, S. Kadyrov, and A. D. Pohl, Escape of mass and entropy for diagonal flows in real rank one situations, Israel J. Math., 210 (2015), 245–295.
M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, The distribution of closed geodesics on the modular surface, and Duke’s theorem, Enseign. Math. (2), 58 (2012), 249–313.
A. Eskin, G. A. Margulis, and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93–141.
A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space, J. Mod. Dyn., 5 (2011), 71–105.
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Springer-Verlag London, Ltd. London, 2011.
U. Hamenstädt, Symbolic dynamics for the Teichmueller flow, arXiv:1112.6107[math.DS].
S. Kadyrov, Positive entropy invariant measures on the space of lattices with escape of mass, Ergodic Theory Dynam. Systems, 32 (2012), 141–157.
S. Kadyrov, Entropy and escape of mass for Hilbert modular spaces, J. Lie Theory, 22 (2012), 701–722.
A. Khintchine, Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), 170–195.
S. Kadyrov and A. D. Pohl, Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and Hausdorff dimension, Ergodic Theory Dynam. Systems, 37 (2017), 539–563.
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.
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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