With cedram.org
Annales Mathématiques
Blaise Pascal
Search for an article
Search within the site
Table of contents for this issue | Previous article | Next article
Georges Grekos; Vladimír Toma; Jana Tomanová
A note on uniform or Banach density
Annales mathématiques Blaise Pascal, 17 no. 1 (2010), p. 153-163, doi: 10.5802/ambp.280
Article PDF | Reviews MR 2674656 | Zbl pre05761664
Class. Math.: 11B05
Keywords: Banach density, uniform density

Résumé - Abstract

In this note we present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers.

Bibliography

[1] Vitaly Bergelson. Sets of recurrence of ${\bf Z}^m$-actions and properties of sets of differences in ${\bf Z}^m$. J. London Math. Soc. (2), 31(2):295-304, 1985. Article |  MR 809951 |  Zbl 0579.10029
[2] Vitaly Bergelson. Ergodic Ramsey theory. Contemporary Math., 65:63-87, 1987.  MR 891243 |  Zbl 0642.10052
[3] Vitaly Bergelson, B. Host and B. Kra. Multiple recurrence and nilsequences. With an appendix by Imre Ruzsa. Invent. Math., 160:261-303, 2005. Article |  MR 2138068 |  Zbl 1087.28007
[4] T. C. Brown and A. R. Freedman. Arithmetic Progressions in Lacunary Sets. Rocky Mountain J. Math., 17:587-596, 1987. Article |  MR 908265 |  Zbl 0632.10052
[5] T. C. Brown and A. R. Freedman. The Uniform Density of Sets of Integers and Fermat’s Last Theorem. C. R. Math. Rep. Acad. Sci. Canada, XII:1-6, 1990.  MR 1043085 |  Zbl 0701.11011
[6] N. G. de Bruijn and P. Erdős. Some linear and some quadratic recursion formulas. I. Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math., 13:374-382, 1951.  MR 47161 |  Zbl 0044.06003
[7] R. E. Dressler and L. Pigno. Small sum sets and the Faber gap condition. Acta Sci. Math. (Szeged), 47:233-237, 1984.  MR 755578 |  Zbl 0564.43006
[8] M. Fekete. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten.. Math. Zeitschr., 17:228-249, 1923. Article |  MR 1544613 |  JFM 49.0047.01
[9] H. Furstenberg. Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, 1981  MR 603625 |  Zbl 0459.28023
[10] Z. Gáliková, B. László and T. Šalát. Remarks on uniform density of sets of integers. Acta Acad. Paed. Agriensis, Sectio Math., 23:3-13, 2002.  MR 1956574 |  Zbl 1012.11012
[11] N. Hegyvári. Note on difference sets in $\mathbb{Z}^n$. Periodica Math. Hungarica, 44:183-185, 2002. Article |  MR 1918685 |  Zbl 1006.11006
[12] R. Jin. Nonstandard methods for upper Banach density problems. Journal of Number Theory, 91:20-38, 2001. Article |  MR 1869316 |  Zbl 1071.11503
[13] R. Nair. On certain solutions of the diophantine equation $x - y = p(z)$. Acta Arithmetica, 62:61-71, 1992. Article |  MR 1179010 |  Zbl 0776.11006
[14] G. Pólya and G. Szegö. Problems and theorems in analysis I. Springer-Verlag, 1972  MR 1492447 |  Zbl 0236.00003
[15] P. Ribenboim. Density results on families of diophantine equations with finitely many solutions. L’Enseignement Mathématique, 39:3-23, 1993.  MR 1225254 |  Zbl 0804.11026
[16] T. Šalát. Remarks on Steinhaus Property and Ratio Sets of Positive Integers. Czech. Math. J., 50:175-183, 2000. Article |  MR 1745470 |  Zbl 1034.11010
[17] T. Šalát and V. Toma. Olivier’s theorem and statistical convergence. Annales Math. Blaise Pascal, 10:305-313, 2003. Cedram |  MR 2031274 |  Zbl 1061.40001
[18] J. Michael Steele. Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), 1997  MR 1422018 |  Zbl 0916.90233