Badly approximable numbers and Littlewood-type problems

Details Badly approximable numbers and Littlewood-type problems Badly approximable numbers and Littlewood-type problems

2009-05-06

arxiv.org/abs/0905.0830v1

Mathematics

Number Theory

14 pages

Scientific paper

We establish that the set of pairs $(\alpha, \beta)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems

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