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On the complexity of a putative counterexample to the p-adic Littlewood conjecture

Badziahin, D.; Bugeaud, Y.; Einsiedler, M.; Kleinbock, D.

On the complexity of a putative counterexample to the p-adic Littlewood conjecture Thumbnail


Authors

D. Badziahin

Y. Bugeaud

M. Einsiedler

D. Kleinbock



Abstract

Let ∥⋅∥ denote the distance to the nearest integer and, for a prime number p, let |⋅|p denote the p-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether infq⩾1q⋅∥qα∥⋅|q|p=0 holds for every badly approximable real number α and every prime number p. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number α grows too rapidly or too slowly, then their conjecture is true for the pair (α,p) with p an arbitrary prime.

Citation

Badziahin, D., Bugeaud, Y., Einsiedler, M., & Kleinbock, D. (2015). On the complexity of a putative counterexample to the p-adic Littlewood conjecture. Compositio Mathematica, 151(09), 1647-1662. https://doi.org/10.1112/s0010437x15007393

Journal Article Type Article
Acceptance Date Jan 15, 2015
Online Publication Date May 19, 2015
Publication Date Sep 1, 2015
Deposit Date Jun 23, 2015
Publicly Available Date Jun 26, 2015
Journal Compositio Mathematica
Print ISSN 0010-437X
Electronic ISSN 1570-5846
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 151
Issue 09
Pages 1647-1662
DOI https://doi.org/10.1112/s0010437x15007393
Keywords Diophantine approximation, Littlewood conjecture, Complexity, Continued fractions, Measure rigidity.
Related Public URLs http://arxiv.org/abs/1405.5545

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