### On continued fraction expansion of potential counterexamples to mixed Littlewood conjecture.

**Badziahin, D ***(Durham University)*

Tuesday 10 June 2014, 14:30-15:30

Seminar Room 2, Newton Institute Gatehouse

#### Abstract

Mixed Littlewood conjecture proposed by de Mathan and Teulie in
2004 states that for every real number $x$ one has
$$
\liminf_{q\to\infty} q\cdot |q|_D\cdot ||qx|| = 0.
$$
where $|*|_D$ is a so called pseudo norm which generalises the standard $p$-adic norm.
In the talk we'll consider the set $\mad$ of potential
counterexamples to this conjecture. Thanks to the results of
Einsiedler and Kleinbock we already know that the Haudorff dimension
of $\mad$ is zero, so this set is very tiny. During the talk we'll
see that the continued fraction expansion of every element in $\mad$
should satisfy some quite restrictive conditions. As one of them
we'll see that for these expansions, considered as infinite words,
the complexity function can neither grow too fast nor too slow.

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