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

Presented by: 
D Badziahin Durham University
Date: 
Tuesday 10th June 2014 - 14:30 to 15:30
Venue: 
INI Seminar Room 2
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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons