Simultaneous Diophantine approximation with a divisibility condition
Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 987-1008.

Dans un article précédent ([9]), nous étudiions des approximations rationnelles simultanées dans ℝ2 qui présentent une certaine analogie avec les fractions continues. Nous obtenions des résultats autour de la conjecture de Littlewood en utilisant de telles approximations. Nous montrons ici que ces résultats restent vrais si l’on ajoute des conditions de divisibilité.

In a previous paper ([9]), we studied certain sequences of simultaneous rational approximations in ℝ2 which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations. Here we show that these results also hold when we add divisibility conditions.

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DOI : 10.5802/jtnb.1303
Classification : 11J13, 11J68
Mots-clés : Littlewood conjecture. Simultaneous Diophantine approximation. Divisibility.
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Simultaneous {Diophantine} approximation with a divisibility condition},
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Bernard de Mathan. Simultaneous Diophantine approximation with a divisibility condition. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 987-1008. doi : 10.5802/jtnb.1303. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1303/

[1] Faustin Adiceam; Erez Nesharim; Fred Lunnon On the t-adic Littlewood conjecture, Duke Math. J., Volume 170 (2021) no. 10, pp. 2371-2419 | MR | Zbl

[2] Dmitry Badziahin; Yann Bugeaud; Manfred Einsiedler; Dmitry Kleinbock On the complexity of a putative counterexample to the p-adic Littlewood conjecture, Compos. Math., Volume 151 (2015) no. 9, pp. 1647-1662 | DOI | MR | Zbl

[3] Alan Baker On an analogue of Littlewood’s Diophantine approximation problem, Mich. Math. J., Volume 11 (1964), pp. 247-250 | MR | Zbl

[4] Yann Bugeaud On the multiples of a badly approximable vector, Acta Arith., Volume 168 (2015) no. 1, pp. 71-81 | DOI | MR | Zbl

[5] J. W. S. Cassels; H. P. F. Swinnerton-Dyer On the product of three homogeneous linear forms and indefinite ternary quadratic forms, Philos. Trans. R. Soc. Lond., Ser. A, Volume 248 (1955), pp. 73-96 | MR | Zbl

[6] Harold Davenport; D. J. Lewis An analogue of a problem of Littlewood, Mich. Math. J., Volume 10 (1963), pp. 157-160 | MR | Zbl

[7] Manfred Einsiedler; Anatole Katok; Elon Lindenstrauss Invariant measures and the set of exceptions to the Littlewood conjecture, Ann. Math. (2), Volume 164 (2006) no. 2, pp. 513-560 | DOI | Zbl

[8] Serge Lang Algebraic numbers, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Group, 1964, ix+163 pages | MR | Zbl

[9] Bernard de Mathan On certain simultaneous rational approximations in ℝ2 and certain rational approximations in ℚp, Monatsh. Math., Volume 200 (2023) no. 4, pp. 849-901 | DOI | MR | Zbl

[10] Bernard de Mathan; Olivier Teulié Problèmes diophantiens simultanés, Monatsh. Math., Volume 143 (2004) no. 3, pp. 229-245 | DOI | MR | Zbl

[11] Leslie G. Peck Simultaneous rational approximations to algebraic numbers, Bull. Am. Math. Soc., Volume 67 (1961), pp. 197-201 | DOI | MR | Zbl

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