Diophantine approximation in Banach spaces

Lior Fishman[1]; David Simmons[2]; Mariusz Urbański[1]

  • [1] University of North Texas Department of Mathematics 1155 Union Circle #311430 Denton, TX 76203-5017, USA
  • [2] Ohio State University Department of Mathematics 231 W. 18th Avenue Columbus Ohio, OH 43210-1174, USA

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 2, page 363-384
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Fishman, Lior, Simmons, David, and Urbański, Mariusz. "Diophantine approximation in Banach spaces." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 363-384. <http://eudml.org/doc/275786>.

@article{Fishman2014,
abstract = {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.},
affiliation = {University of North Texas Department of Mathematics 1155 Union Circle #311430 Denton, TX 76203-5017, USA; Ohio State University Department of Mathematics 231 W. 18th Avenue Columbus Ohio, OH 43210-1174, USA; University of North Texas Department of Mathematics 1155 Union Circle #311430 Denton, TX 76203-5017, USA},
author = {Fishman, Lior, Simmons, David, Urbański, Mariusz},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {10},
number = {2},
pages = {363-384},
publisher = {Société Arithmétique de Bordeaux},
title = {Diophantine approximation in Banach spaces},
url = {http://eudml.org/doc/275786},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Fishman, Lior
AU - Simmons, David
AU - Urbański, Mariusz
TI - Diophantine approximation in Banach spaces
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/10//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 2
SP - 363
EP - 384
AB - 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.
LA - eng
UR - http://eudml.org/doc/275786
ER -

References

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  1. F. D. Ancel, T. Dobrowolski, and J. Grabowski, Closed subgroups in Banach spaces, Studia Math. 109 (1994), no. 3, 277–290. Zbl0840.46012MR1274013
  2. J. Chaika, Y. Cheung, and H. A. Masur, Winning games for bounded geodesics in moduli spaces of quadratic differentials,http://arxiv.org/abs/1109.5976 preprint 2011. Zbl1284.30034
  3. M. M. Dodson and B. Everitt, Metrical Diophantine approximation for quaternions, http://arxiv.org/abs/1109.3229, 2012, preprint. Zbl06399691MR3286521
  4. L. Fishman, D. Y. Kleinbock, K. Merrill, and D. S. Simmons, Intrinsic Diophantine approximation on manifolds, http://arxiv.org/abs/1405.7650 preprint 2014. 
  5. L. Fishman and D. S. Simmons, Intrinsic approximation for fractals defined by rational iterated function systems - Mahler’s research suggestion, http://arxiv.org/abs/1208.2089 preprint 2012, to appear in Proc. Lond. Math. Soc. (3). Zbl1309.11059MR3237740
  6. L. Fishman and D. S. Simmons, Unconventional height functions in simultaneous Diophantine approximation, http://arxiv.org/abs/1401.8266 preprint 2014. MR3237740
  7. L. Fishman, D. S. Simmons, and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces, http://arxiv.org/abs/1301.5630, 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. Zbl0763.28009MR1161274
  10. S. Kristensen, On well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 255–268. Zbl1088.11056MR2006063

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