Badly approximable numbers and vectors in Cantor-like sets

Dani, S.; Shah, Hemangi

doi:10.1090/S0002-9939-2011-11105-5

Badly approximable numbers and vectors in Cantor-like sets
HTML articles powered by AMS MathViewer

by S. G. Dani and Hemangi Shah PDF

Proc. Amer. Math. Soc. 140 (2012), 2575-2587 Request permission

Abstract:

We show that a large class of Cantor-like sets of ${{\mathbb R}}^{d}, d \geq 1$, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when $d\geq 2$. An analogous result is also proved for subsets of ${\mathbb R}^d$ arising in the study of geodesic flows corresponding to $(d+1)$-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in ${\mathbb R}$. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets.

References

C. S. Aravinda and Enrico Leuzinger, Bounded geodesics in rank-$1$ locally symmetric spaces, Ergodic Theory Dynam. Systems 15 (1995), no. 5, 813–820. MR 1356615, DOI 10.1017/S0143385700009640
M. Bachir Bekka and Matthias Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000. MR 1781937, DOI 10.1017/CBO9780511758898
R. Broderick, L. Fishman and D. Kleinbock, Schmidt’s game, fractals, and orbits of toral endomorphisms, Ergod. Th. Dynam. Sys. 31 (2011), 1095–1117.
S.G. Dani, Bounded geodesics on manifolds of negative curvature, preprint, 1986 (unpublished).
S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv. 61 (1986), no. 4, 636–660. MR 870710, DOI 10.1007/BF02621936
S. G. Dani, On badly approximable numbers, Schmidt games and bounded orbits of flows, Number theory and dynamical systems (York, 1987) London Math. Soc. Lecture Note Ser., vol. 134, Cambridge Univ. Press, Cambridge, 1989, pp. 69–86. MR 1043706, DOI 10.1017/CBO9780511661983.006
S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 523–529. MR 980795, DOI 10.1017/S0143385700004673
Lior Fishman, Schmidt’s game on fractals, Israel J. Math. 171 (2009), 77–92. MR 2520102, DOI 10.1007/s11856-009-0041-x
Lior Fishman, Schmidt’s game, badly approximable matrices and fractals, J. Number Theory 129 (2009), no. 9, 2133–2153. MR 2528057, DOI 10.1016/j.jnt.2009.02.005
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
D. Kleinbock and B. Weiss, Modified Schmidt games and a conjecture of Margulis, preprint, 2010. arXiv:1001.5017vl [math.DS] 27 Jan 2010
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
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