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ON MULTIPLICATIVELY BADLY APPROXIMABLE NUMBERS

  • Dzmitry A. Badziahin (a1)

Abstract

The Littlewood conjecture states that for all (α,β)∈ℝ2. We show that with the additional factor of log q⋅log log q, the statement is false. Indeed, our main result implies that the set of (α,β) for which is of full dimension.

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[1]Badziahin, D., Pollington, A. and Velani, S., On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture. Ann. of Math. (2) 174 (2011), 18371883.
[2]Badziahin, D. and Velani, S., Multiplicatively badly approximable numbers and generalised Cantor sets. Adv. Math. 228(5) (2011), 27662796.
[3]Bugeaud, Y. and Moshchevitin, N., Badly approximable numbers and Littlewood-type problems. Math. Proc. Cambridge Philos. Soc. 150 (2011), 215226.
[4]Einsiedler, M., Katok, A. and Lindenstrauss, E., Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164 (2006), 513560.
[5]Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons (Chichester, 1990).
[6]Pollington, A. and Velani, S., On a problem in simultaneously Diophantine approximation: Littlewood’s conjecture. Acta Math. 66 (2000), 2940.
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