Diophantine approximation in Banach spaces
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 363-384.

Dans cet article, nous étendons à la dimension infinie la théorie de l’approximation diophantienne simultanée. De plus, nous discutons des théorèmes de type Dirichlet dans un cadre très général et nous définissons ce que signifie étre optimal pour un tel théorème. Nous montrons que l’optimalité est impliquée par, mais n’implique pas, l’existence de points mal approchables.

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

Reçu le : 2013-03-11
Accepté le : 2013-05-13
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.871
@article{JTNB_2014__26_2_363_0,
     author = {Lior Fishman and David Simmons and Mariusz Urba\'nski},
     title = {Diophantine approximation in Banach spaces},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {363--384},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.871},
     mrnumber = {3320484},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_363_0/}
}
Lior Fishman; David Simmons; Mariusz Urbański. Diophantine approximation in Banach spaces. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 363-384. doi : 10.5802/jtnb.871. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_363_0/

[1] F. D. Ancel, T. Dobrowolski, and J. Grabowski, Closed subgroups in Banach spaces, Studia Math. 109 (1994), no. 3, 277–290. | MR 1274013 | Zbl 0840.46012

[2] J. Chaika, Y. Cheung, and H. A. Masur, Winning games for bounded geodesics in moduli spaces of quadratic differentials, preprint 2011.

[3] M. M. Dodson and B. Everitt, Metrical Diophantine approximation for quaternions, , 2012, preprint. | MR 3286521

[4] L. Fishman, D. Y. Kleinbock, K. Merrill, and D. S. Simmons, Intrinsic Diophantine approximation on manifolds, preprint 2014.

[5] L. Fishman and D. S. Simmons, Intrinsic approximation for fractals defined by rational iterated function systems - Mahler’s research suggestion, preprint 2012, to appear in Proc. Lond. Math. Soc. (3). | MR 3237740

[6] L. Fishman and D. S. Simmons, Unconventional height functions in simultaneous Diophantine approximation, preprint 2014. | MR 3237740

[7] L. Fishman, D. S. Simmons, and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, , 2013, preprint.

[8] G. H. Hardy, Orders of infinity. The Infinitärcalcül of Paul du Bois-Reymond, Cambridge Tracts in Mathematics and Mathematical Physics, No. 12, Hafner Publishing Co., New York, 1971.

[9] B. R. Hunt, T. D. Sauer, and J. A. Yorke, Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. | MR 1161274 | Zbl 0763.28009

[10] S. Kristensen, On well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 255–268. | MR 2006063 | Zbl 1088.11056