On a mixed Littlewood conjecture for quadratic numbers
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 207-215.

Nous étudions un problème diophantien simultané relié à la conjecture de Littlewood. En utilisant des minorations connues de formes linéaires de logarithmes p-adiques, nous montrons qu’un résultat que nous avons précédemment obtenu, concernant les nombres quadratiques, est presque optimal.

We study a simultaneous diophantine problem related to Littlewood’s conjecture. Using known estimates for linear forms in p-adic logarithms, we prove that a previous result, concerning the particular case of quadratic numbers, is close to be the best possible.

@article{JTNB_2005__17_1_207_0,
     author = {Bernard de Mathan},
     title = {On a mixed Littlewood conjecture for quadratic numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {207--215},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.487},
     zbl = {1165.11325},
     mrnumber = {2152221},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.487/}
}
Bernard de Mathan. On a mixed Littlewood conjecture for quadratic numbers. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 207-215. doi : 10.5802/jtnb.487. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.487/

[1] M. Bauer, M. Bennett, Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270. | MR 1908198 | Zbl 1010.11020

[2] Y. Bugeaud, M. Laurent, Minoration effective de la distance p-adique entre puissances de nombres algébriques. J. Number Theory 61 (1996), 311–342. | MR 1423057 | Zbl 0870.11045

[3] B. de Mathan, Linear forms in logarithms and simultaneous Diophantine approximation. (To appear).

[4] B. de Mathan, Approximations diophantiennes dans un corps local. Bull. Soc. math. France, Mémoire 21 (1970). | Numdam | MR 274396 | Zbl 0221.10037

[5] B. de Mathan, O. Teulié, Problèmes diophantiens simultanés. Monatshefte Math. 143 (2004), 229–245. | MR 2103807 | Zbl 02132191

[6] D. Ridout, Rational approximations to algebraic numbers. Mathematika 4 (1957), 125–131. | MR 93508 | Zbl 0079.27401

[7] L. G. Peck, Simultaneous rational approximations to algebraic numbers. Bull. Amer. Math. Soc. 67 (1961), 197–201. | MR 122772 | Zbl 0098.26302

[8] K. Yu, p-adic logarithmic forms and group varieties II. Acta Arith. 89 (1999), 337–378. | MR 1703864 | Zbl 0928.11031