An extension of quantitative nondivergence and applications to Diophantine exponents

Kleinbock, Dmitry

doi:10.1090/S0002-9947-08-04592-3

An extension of quantitative nondivergence and applications to Diophantine exponents
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Trans. Amer. Math. Soc. 360 (2008), 6497-6523 Request permission

Abstract:

We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of $\mathbb {R}^{n}$ and their nondegenerate submanifolds.

References

V. Beresnevich, A Groshev type theorem for convergence on manifolds, Acta Math. Hungar. 94 (2002), no. 1-2, 99–130. MR 1905790, DOI 10.1023/A:1015662722298
V. V. Berasnevīch, V. Ī. Bernīk, Kh. Dykīnsan, and M. Dodsan, On linear manifolds for which the Khinchin approximation theorem holds, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk 2 (2000), 14–17, 139 (Belorussian, with English and Russian summaries). MR 1820985
V. V. Beresnevich, V. I. Bernik, D. Y. Kleinbock, and G. A. Margulis, Metric Diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds, Mosc. Math. J. 2 (2002), no. 2, 203–225. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944505, DOI 10.17323/1609-4514-2002-2-2-203-225
V. I. Bernik and M. M. Dodson, Metric Diophantine approximation on manifolds, Cambridge Tracts in Mathematics, vol. 137, Cambridge University Press, Cambridge, 1999. MR 1727177, DOI 10.1017/CBO9780511565991
V. Bernik, D. Kleinbock, and G. A. Margulis, Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions, Internat. Math. Res. Notices 9 (2001), 453–486. MR 1829381, DOI 10.1155/S1073792801000241
J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708
S. G. Dani, On invariant measures, minimal sets and a lemma of Margulis, Invent. Math. 51 (1979), no. 3, 239–260. MR 530631, DOI 10.1007/BF01389917
S. G. Dani, On orbits of unipotent flows on homogeneous spaces, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 25–34. MR 758891, DOI 10.1017/S0143385700002248
S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55–89. MR 794799, DOI 10.1515/crll.1985.359.55
S. G. Dani, On orbits of unipotent flows on homogeneous spaces. II, Ergodic Theory Dynam. Systems 6 (1986), no. 2, 167–182. MR 857195, DOI 10.1017/s0143385700003382
M. M. Dodson, Hausdorff dimension, lower order and Khintchine’s theorem in metric Diophantine approximation, J. Reine Angew. Math. 432 (1992), 69–76. MR 1184759, DOI 10.1515/crll.1992.432.69
Anish Ghosh, A Khintchine-type theorem for hyperplanes, J. London Math. Soc. (2) 72 (2005), no. 2, 293–304. MR 2156655, DOI 10.1112/S0024610705006587
Anish Ghosh, Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory 124 (2007), no. 2, 454–469. MR 2321374, DOI 10.1016/j.jnt.2006.10.009
—, Dynamics on homogeneous spaces and Diophantine approximation on manifolds, Ph. D. Thesis, Brandeis University, 2006.
Dmitry Kleinbock, Some applications of homogeneous dynamics to number theory, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 639–660. MR 1858548, DOI 10.1090/pspum/069/1858548
D. Kleinbock, Extremal subspaces and their submanifolds, Geom. Funct. Anal. 13 (2003), no. 2, 437–466. MR 1982150, DOI 10.1007/s000390300011
—, Baker-Sprindžuk conjectures for complex analytic manifolds, in: Algebraic groups and Arithmetic, TIFR, India, 2004, pp. 539-553.
Dmitry Kleinbock, Elon Lindenstrauss, and Barak Weiss, On fractal measures and Diophantine approximation, Selecta Math. (N.S.) 10 (2004), no. 4, 479–523. MR 2134453, DOI 10.1007/s00029-004-0378-2
D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148 (1998), no. 1, 339–360. MR 1652916, DOI 10.2307/120997
Dmitry Kleinbock and George Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv. 82 (2007), no. 3, 519–581. MR 2314053, DOI 10.4171/CMH/102
Dmitry Kleinbock and Barak Weiss, Badly approximable vectors on fractals, Israel J. Math. 149 (2005), 137–170. Probability in mathematics. MR 2191212, DOI 10.1007/BF02772538
Dmitry Kleinbock and Barak Weiss, Friendly measures, homogeneous flows and singular vectors, Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, pp. 281–292. MR 2180240, DOI 10.1090/conm/385/07201
Dmitry Kleinbock and Barak Weiss, Dirichlet’s theorem on Diophantine approximation and homogeneous flows, J. Mod. Dyn. 2 (2008), no. 1, 43–62. MR 2366229, DOI 10.3934/jmd.2008.2.43
G. A. Margulis, On the action of unipotent groups in the space of lattices, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 365–370. MR 0470140
Gregory Margulis, Diophantine approximation, lattices and flows on homogeneous spaces, A panorama of number theory or the view from Baker’s garden (Zürich, 1999) Cambridge Univ. Press, Cambridge, 2002, pp. 280–310. MR 1975458, DOI 10.1017/CBO9780511542961.019
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Andrew Pollington and Sanju L. Velani, Metric Diophantine approximation and “absolutely friendly” measures, Selecta Math. (N.S.) 11 (2005), no. 2, 297–307. MR 2183850, DOI 10.1007/s00029-005-0007-8
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234
Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, DOI 10.1215/S0012-7094-91-06311-8
M. Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994), no. 2, 236–257. MR 1262705, DOI 10.1007/BF01895839
Wolfgang M. Schmidt, Diophantine approximation and certain sequences of lattices, Acta Arith. 18 (1971), 195–178. (errata insert). MR 286751, DOI 10.4064/aa-18-1-165-178
Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
V. G. Sprindžuk, Achievements and problems of the theory of Diophantine approximations, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 3–68, 248 (Russian). MR 586190
Bernd O. Stratmann and Mariusz Urbański, Diophantine extremality of the Patterson measure, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 297–304. MR 2212281, DOI 10.1017/S0305004105009114
Mariusz Urbański, Diophantine approximation and self-conformal measures, J. Number Theory 110 (2005), no. 2, 219–235. MR 2122607, DOI 10.1016/j.jnt.2004.07.004