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arXiv · 1101.1855

On multiplicatively badly approximable numbers

Dzmitry Badziahin
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Abstract

The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that the set of (x,y) for which liminf_{q\to\infty} q . log q . loglog q . ||qx|| . ||qy|| > 0 is of full dimension.

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Dzmitry Badziahin. 2011-01-10. On multiplicatively badly approximable numbers. https://doi.org/10.1112/s0025579312000095

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