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Measure rigidity and $p$-adic Littlewood-type problems

Published online by Cambridge University Press:  04 December 2007

Manfred Einsiedler  and
Dmitry Kleinbock
Manfred Einsiedler
Affiliation:
Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA einsiedler.1@osu.edu
Dmitry Kleinbock
Affiliation:
Department of Mathematics, MS 050, Brandeis University, PO Box 9110, Waltham, MA 02454, USA kleinboc@brandeis.edu
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Abstract

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Various $p$-adic versions of Littlewood's conjecture are investigated, generalizing a set-up considered recently by de Mathan and Teulié. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss.

Keywords

invariant measuresrigidityhigher rank abelian actionsDiophantine approximation$p$-adicLittlewood's conjecture11J04 (primary)11J8337D4037A35 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 3 , May 2007 , pp. 689 - 702
Copyright
Foundation Compositio Mathematica 2007