Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud

doi:10.5802/pmb.1

Yann Bugeaud ¹

¹ Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 5-18.

Abstract
Résumé

The Littlewood conjecture in Diophantine approximation claims that

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0

holds for all real numbers $α$ and $β$ , where $∥ \cdot ∥$ denotes the distance to the nearest integer. Its $p$ -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0

holds for every real number $α$ and every prime number $p$ , where ${| \cdot |}_{p}$ denotes the $p$ -adic absolute value normalized by ${| p |}_{p} = p^{- 1}$ . We survey the known results on these conjectures and highlight recent developments.

En approximation diophantienne, la conjecture de Littlewood stipule que tous les nombres réels $α$ et $β$ vérifient

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0,

où $∥ \cdot ∥$ désigne la distance à l’entier le plus proche. Son analogue $p$ -adique, formulé par de Mathan et Teulié en 2004, affirme que l’égalité

inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0

est valable pour tout nombre réel $α$ et tout nombre premier $p$ , où ${| \cdot |}_{p}$ est la valeur absolue $p$ -adique normalisée par ${| p |}_{p} = p^{- 1}$ . Nous donnons un survol des résultats connus sur ces conjectures en insistant sur les développements récents.

Received: 2014-05-24
Published online: 2015-04-13

Zbl

DOI: 10.5802/pmb.1

Classification: 11J04, 11J13, 11J61
Keywords: Simultaneous approximation, Littlewood conjecture

Author's affiliations:

Yann Bugeaud ¹

¹ Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France

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Yann Bugeaud. Around the Littlewood conjecture in Diophantine approximation. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 5-18. doi : 10.5802/pmb.1. https://pmb.centre-mersenne.org/articles/10.5802/pmb.1/

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