The Littlewood conjecture in Diophantine approximation claims that
holds for all real numbers and , where denotes the distance to the nearest integer. Its -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that
holds for every real number and every prime number , where denotes the -adic absolute value normalized by . 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 et vérifient
où désigne la distance à l’entier le plus proche. Son analogue -adique, formulé par de Mathan et Teulié en 2004, affirme que l’égalité
est valable pour tout nombre réel et tout nombre premier , où est la valeur absolue -adique normalisée par . Nous donnons un survol des résultats connus sur ces conjectures en insistant sur les développements récents.
Published online:
DOI: 10.5802/pmb.1
Keywords: Simultaneous approximation, Littlewood conjecture
Yann Bugeaud 1
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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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