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THE MIXED SCHMIDT CONJECTURE IN THE THEORY OF DIOPHANTINE APPROXIMATION

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

Abstract

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.
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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