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

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

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.
• Public URL: https://durham-repository.worktribe.com/output/1405804
• Related Public URLs: http://arxiv.org/abs/1405.5545