Multiplicatively badly approximable numbers and generalised Cantor sets

By Dzmitry Badziahin, Sanju Velani. 2010

47 citations recorded by the source corpus.

Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminf_{q \to \infty} q . |q|_p . ||q x|| = 0 for all real numbers x. 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 for which liminf_{q\to\infty} q . log q . loglog q. |q|_p . ||qx|| > 0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/056ce00e8e35dec64c83741f839f7871983a7fe6

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Citations in the corpus

Metric Diophantine Approximation : aspects of recent work (2016, methodology): “Building upon the one-dimensional, generalised Cantor sets framework formulated in [7], an abstract ‘metric space’ framework of higher dimensional generalised Cantor sets, branded as ‘Cantor winning sets’, has recently been introduced in [5].”
Badly approximable points on manifolds (2013, methodology): “Hence Theorem 5 above is an immediate consequence of Theorem 4 from [16].”