On multiplicatively badly approximable numbers

Details On multiplicatively badly approximable numbers On multiplicatively badly approximable numbers

2011-01-10

arxiv.org/abs/1101.1855v1

Mathematics

Number Theory

22 pages

Scientific paper

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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