Intrinsic Diophantine approximation on manifolds: General theory

Fishman, Lior; Kleinbock, Dmitry; Merrill, Keith; Simmons, David

doi:10.1090/tran/6971

Intrinsic Diophantine approximation on manifolds: General theory
HTML articles powered by AMS MathViewer

by Lior Fishman, Dmitry Kleinbock, Keith Merrill and David Simmons PDF

Trans. Amer. Math. Soc. 370 (2018), 577-599 Request permission

Abstract:

We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the “intrinsic Dirichlet exponent” for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on $M$ distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the simplex lemma for rationals lying on $M$ which provides new insights on the local distribution of rational points on nondegenerate manifolds.

References

Jinpeng An, Lifan Guan, and Dmitry Kleinbock, Bounded orbits of diagonalizable flows on $\mathrm {SL}_3(\Bbb {R})/\mathrm {SL}_3(\Bbb {Z})$, Int. Math. Res. Not. IMRN 24 (2015), 13623–13652. MR 3436158, DOI 10.1093/imrn/rnv120
Dzmitry Badziahin and Sanju Velani, Badly approximable points on planar curves and a problem of Davenport, Math. Ann. 359 (2014), no. 3-4, 969–1023. MR 3231023, DOI 10.1007/s00208-014-1020-z
Victor Beresnevich, Rational points near manifolds and metric Diophantine approximation, Ann. of Math. (2) 175 (2012), no. 1, 187–235. MR 2874641, DOI 10.4007/annals.2012.175.1.5
Victor Beresnevich, Badly approximable points on manifolds, Invent. Math. 202 (2015), no. 3, 1199–1240. MR 3425389, DOI 10.1007/s00222-015-0586-8
Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, and Barak Weiss, The set of badly approximable vectors is strongly $C^1$ incompressible, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 319–339. MR 2981929, DOI 10.1017/S0305004112000242
Ryan Broderick, Lior Fishman, and Asaf Reich, Intrinsic approximation on Cantor-like sets, a problem of Mahler, Mosc. J. Comb. Number Theory 1 (2011), no. 4, 3–12. MR 2988380
Ryan Broderick, Lior Fishman, and David Simmons, Badly approximable systems of affine forms and incompressibility on fractals, J. Number Theory 133 (2013), no. 7, 2186–2205. MR 3035957, DOI 10.1016/j.jnt.2012.12.004
Natalia Budarina, Detta Dickinson, and Jason Levesley, Simultaneous Diophantine approximation on polynomial curves, Mathematika 56 (2010), no. 1, 77–85. MR 2604984, DOI 10.1112/S0025579309000382
Lior Fishman, Dmitry Kleinbock, Keith Merrill, and David Simmons, Intrinsic Diophantine approximation on quadric hypersurfaces, http://arxiv.org/abs/1405.7650, preprint 2015.
Lior Fishman, Keith Merrill, and David Simmons, Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces, http://arxiv.org/abs/1502.07648, preprint 2015.
Lior Fishman and David Simmons, Intrinsic approximation for fractals defined by rational iterated function systems: Mahler’s research suggestion, Proc. Lond. Math. Soc. (3) 109 (2014), no. 1, 189–212. MR 3237740, DOI 10.1112/plms/pdu002
Lior Fishman and David Simmons, Extrinsic Diophantine approximation on manifolds and fractals, J. Math. Pures Appl. (9) 104 (2015), no. 1, 83–101 (English, with English and French summaries). MR 3350721, DOI 10.1016/j.matpur.2015.02.002
Lior Fishman, David Simmons, and Mariusz Urbański, Diophantine approximation in Banach spaces, J. Théor. Nombres Bordeaux 26 (2014), no. 2, 363–384 (English, with English and French summaries). MR 3320484, DOI 10.5802/jtnb.871
Lior Fishman, David Simmons, and Mariusz Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, http://arxiv.org/abs/1301.5630, preprint 2013, to appear Mem. Amer. Math. Soc.
Anish Ghosh, Alexander Gorodnik, and Amos Nevo, Diophantine approximation and automorphic spectrum, Int. Math. Res. Not. IMRN 21 (2013), 5002–5058. MR 3123673, DOI 10.1093/imrn/rns198
Anish Ghosh, Alexander Gorodnik, and Amos Nevo, Metric Diophantine approximation on homogeneous varieties, Compos. Math. 150 (2014), no. 8, 1435–1456. MR 3252026, DOI 10.1112/S0010437X13007859
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
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
Dmitry Kleinbock and Tue Ly, Badly approximable $S$-numbers and absolute Schmidt games, J. Number Theory 164 (2016), 13–42. MR 3474376, DOI 10.1016/j.jnt.2015.12.014
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 Barak Weiss, Modified Schmidt games and Diophantine approximation with weights, Adv. Math. 223 (2010), no. 4, 1276–1298. MR 2581371, DOI 10.1016/j.aim.2009.09.018
Dmitry Kleinbock and Barak Weiss, Values of binary quadratic forms at integer points and Schmidt games, Recent trends in ergodic theory and dynamical systems, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 77–92. MR 3330339, DOI 10.1090/conm/631/12597
Dmitry Kleinbock and Keith Merrill, Rational approximation on spheres, Israel J. Math. 209 (2015), no. 1, 293–322. MR 3430242, DOI 10.1007/s11856-015-1219-z
Adrien-Marie Legendre, Essai sur la théorie des nombres, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009 (French). Reprint of the second (1808) edition. MR 2859036, DOI 10.1017/CBO9780511693199
Curtis T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal. 20 (2010), no. 3, 726–740. MR 2720230, DOI 10.1007/s00039-010-0078-3
Erez Nesharim and David Simmons, $\textbf {Bad} (s,t)$ is hyperplane absolute winning, Acta Arith. 164 (2014), no. 2, 145–152. MR 3224831, DOI 10.4064/aa164-2-4
Oskar Perron, Über diophantische Approximationen, Math. Ann. 83 (1921), no. 1-2, 77–84 (German). MR 1512000, DOI 10.1007/BF01464229
Wolfgang M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178–199. MR 195595, DOI 10.1090/S0002-9947-1966-0195595-4
Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
Eric Schmutz, Rational points on the unit sphere, Cent. Eur. J. Math. 6 (2008), no. 3, 482–487. MR 2425007, DOI 10.2478/s11533-008-0038-4
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
K. Wolsson, Linear dependence of a function set of $m$ variables with vanishing generalized Wronskians, Linear Algebra Appl. 117 (1989), 73–80. MR 993032, DOI 10.1016/0024-3795(89)90548-X