On multiplicatively badly approximable numbers

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.

22 pages

Cited by in corpus (8)

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• Sums of Reciprocals of Fractional Parts over Aligned Boxes
• Geometric considerations around the Littlewood conjecture
• Multiplicatively badly approximable matrices up to logarithmic factors
• Some refined results on mixed Littlewood conjecture for pseudo-absolute values