Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud[1]

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

Publications mathématiques de Besançon (2014)

  • Issue: 1, page 5-18
  • ISSN: 1958-7236

Abstract

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The Littlewood conjecture in Diophantine approximation claims that inf q ≥ 1 q · ∥ q α ∥ · ∥ q β ∥ = 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 inf q ≥ 1 q · ∥ q α ∥ · | 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.

How to cite

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Bugeaud, Yann. "Around the Littlewood conjecture in Diophantine approximation." Publications mathématiques de Besançon (2014): 5-18. <http://eudml.org/doc/275760>.

@article{Bugeaud2014,
abstract = {The Littlewood conjecture in Diophantine approximation claims that\[ \inf \_\{q \ge 1\} \, q \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert = 0 \]holds for all real numbers $\alpha $ and $\beta $, where $\Vert \cdot \Vert $ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that\[ \inf \_\{q \ge 1\} \, q \cdot \Vert q \alpha \Vert \cdot \vert q \vert \_p = 0 \]holds for every real number $\alpha $ 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.},
affiliation = {Mathématiques Université de Strasbourg 7, rue René Descartes, F-67084 Strasbourg France},
author = {Bugeaud, Yann},
journal = {Publications mathématiques de Besançon},
keywords = {Simultaneous approximation; Littlewood conjecture; simultaneous approximation},
language = {eng},
number = {1},
pages = {5-18},
publisher = {Presses universitaires de Franche-Comté},
title = {Around the Littlewood conjecture in Diophantine approximation},
url = {http://eudml.org/doc/275760},
year = {2014},
}

TY - JOUR
AU - Bugeaud, Yann
TI - Around the Littlewood conjecture in Diophantine approximation
JO - Publications mathématiques de Besançon
PY - 2014
PB - Presses universitaires de Franche-Comté
IS - 1
SP - 5
EP - 18
AB - The Littlewood conjecture in Diophantine approximation claims that\[ \inf _{q \ge 1} \, q \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert = 0 \]holds for all real numbers $\alpha $ and $\beta $, where $\Vert \cdot \Vert $ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that\[ \inf _{q \ge 1} \, q \cdot \Vert q \alpha \Vert \cdot \vert q \vert _p = 0 \]holds for every real number $\alpha $ 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.
LA - eng
KW - Simultaneous approximation; Littlewood conjecture; simultaneous approximation
UR - http://eudml.org/doc/275760
ER -

References

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