On multiplicatively badly approximable numbers.

@article{Badziahin2011OnMB,
  title={On multiplicatively badly approximable numbers.},
  author={Dzmitry Badziahin},
  journal={Mathematika},
  year={2011},
  volume={59},
  pages={31-55}
}
  • Dzmitry Badziahin
  • Published 2011
  • Mathematics
  • Mathematika
  • 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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