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  • Dzmitry Badziahin (a1), Jason Levesley (a2) and Sanju Velani (a3)


Let 𝒟=(dn)n=1 be a sequence of integers with dn≥2, and let (i,j) be a pair of strictly positive numbers with i+j=1. We prove that the set of x∈ℝ for which there exists some constant c(x)≧0 such that is one-quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. This, in turn, establishes the natural analogue of Schmidt’s conjecture within the framework of the de Mathan–Teulié conjecture, also known as the “mixed Littlewood conjecture”.



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[1]Badziahin, D., Pollington, A. and Velani, S., On a problem in simultaneously Diophantine approximation: Schmidt’s conjecture. Ann. of Math. (2) (to appear). Preprint, 2010, arXiv:1001.2694.
[2]Bugeaud, Y., Drmota, M. and de Mathan, B., On a mixed Littlewood conjecture in Diophantine approximation. Acta Arith. 128(2) (2007), 107124; MR 2313997(2008d:11067).
[3]Bugeaud, Y., Haynes, A. and Velani, S., Metric considerations concerning the mixed Littlewood conjecture. Int. J. Number Theory (to appear). Preprint, 2009, arXiv:0909.3923v1.
[4]Bugeaud, Y. and Moshchevitin, N., Badly approximable numbers and Littlewood-type problems. Math. Proc. Cambridge Philos. Soc. 150 (2011), 215226.
[5]Einsiedler, M., Katok, A. and Lindenstrauss, E., Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164(2) (2006), 513560; MR 2247967(2007j:22032).
[6]Einsiedler, M. and Kleinbock, D., Measure rigidity and p-adic Littlewood-type problems. Compositio Math. 143(3) (2007), 689702; MR 2330443(2008f:11076).
[7]Harrap, S. and Yusupova, T., On a mixed Khintchine problem in Diophantine approximation, in preparation.
[8]de Mathan, B. and Teulié, O., Problèmes diophantiens simultanés. Monatsh. Math. 143(3) (2004), 229245; MR 2103807(2005h:11147).
[9]Pollington, A. and Velani, S., On a problem in simultaneously Diophantine approximation: Littlewood’s conjecture. Acta Math. 66 (2000), 2940; MR 1819996(2002a:11076).
[10]Schmidt, W. M., On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178199; MR 0195595(33#3793).
[11]Schmidt, W. M., Open problems in Diophantine approximation. In Approximations diophantiennes et nombres transcendants (Colloque de Luminy, 1982) (Progress in Mathematics 31), Birkhäuser (Boston, 1983), 271287.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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