Around the Littlewood conjecture in Diophantine approximation
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 5-18.

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

infq1q·qα·qβ=0,

· 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

infq1q·qα·|q|p=0

est valable pour tout nombre réel α et tout nombre premier p, où |·| 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.

The Littlewood conjecture in Diophantine approximation claims that

infq1q·qα·qβ=0

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

infq1q·qα·|q|p=0

holds for every real number α and every prime number p, where |·| 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.

Reçu le :
Publié le :
DOI : 10.5802/pmb.1
Classification : 11J04, 11J13, 11J61
Mots clés : Simultaneous approximation, Littlewood conjecture
Bugeaud, Yann 1

1 Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France
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Bugeaud, Yann. 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. http://www.numdam.org/articles/10.5802/pmb.1/

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