Annales Mathématiques
Blaise Pascal
no. 1
A note on uniform or Banach density
Georges Grekos1 ; Vladimír Toma2 ; Jana Tomanová3
1 Department of Mathematics Université de Saint-Etienne 23, rue du Docteur Paul Michelon F-42023 Saint-Etienne Cédex 2 France
2 Department of Mathematical and Numerical Analysis Comenius University Mlynská dolina 842 48 Bratislava Slovakia
3 Department of Algebra and Number Theory Comenius University Mlynská dolina 842 48 Bratislava Slovakia
Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 153-163.

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

DOI : 10.5802/ambp.280
Classification : 11B05
Mots clés : Banach density, uniform density

Georges Grekos 1 ; Vladimír Toma 2 ; Jana Tomanová 3

1 Department of Mathematics Université de Saint-Etienne 23, rue du Docteur Paul Michelon F-42023 Saint-Etienne Cédex 2 France
2 Department of Mathematical and Numerical Analysis Comenius University Mlynská dolina 842 48 Bratislava Slovakia
3 Department of Algebra and Number Theory Comenius University Mlynská dolina 842 48 Bratislava Slovakia 
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Georges Grekos; Vladimír Toma; Jana Tomanová. A note on uniform or Banach density. Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 153-163. doi : 10.5802/ambp.280. https://ambp.centre-mersenne.org/articles/10.5802/ambp.280/

[1] Vitaly Bergelson Sets of recurrence of Z m -actions and properties of sets of differences in Z m , J. London Math. Soc. (2), Volume 31 (1985) no. 2, pp. 295-304 | DOI | MR | Zbl

[2] Vitaly Bergelson Ergodic Ramsey theory, Contemporary Math., Volume 65 (1987), pp. 63-87 | MR | Zbl

[3] Vitaly Bergelson; B. Host; B. Kra Multiple recurrence and nilsequences. With an appendix by Imre Ruzsa, Invent. Math., Volume 160 (2005), pp. 261-303 | DOI | MR | Zbl

[4] T. C. Brown; A. R. Freedman Arithmetic Progressions in Lacunary Sets, Rocky Mountain J. Math., Volume 17 (1987), pp. 587-596 | DOI | MR | Zbl

[5] T. C. Brown; A. R. Freedman The Uniform Density of Sets of Integers and Fermat’s Last Theorem, C. R. Math. Rep. Acad. Sci. Canada, Volume XII (1990), pp. 1-6 | MR | Zbl

[6] N. G. de Bruijn; P. Erdős Some linear and some quadratic recursion formulas, I. Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math., Volume 13 (1951), pp. 374-382 | MR | Zbl

[7] R. E. Dressler; L. Pigno Small sum sets and the Faber gap condition, Acta Sci. Math. (Szeged), Volume 47 (1984), pp. 233-237 | MR | Zbl

[8] M. Fekete Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten., Math. Zeitschr., Volume 17 (1923), pp. 228-249 | DOI | MR

[9] H. Furstenberg Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981 (M. B. Porter Lectures) | MR | Zbl

[10] Z. Gáliková; B. László; T. Šalát Remarks on uniform density of sets of integers, Acta Acad. Paed. Agriensis, Sectio Math., Volume 23 (2002), pp. 3-13 | MR | Zbl

[11] N. Hegyvári Note on difference sets in ℤ n , Periodica Math. Hungarica, Volume 44 (2002), pp. 183-185 | DOI | MR | Zbl

[12] R. Jin Nonstandard methods for upper Banach density problems, Journal of Number Theory, Volume 91 (2001), pp. 20-38 | DOI | MR | Zbl

[13] R. Nair On certain solutions of the diophantine equation x-y=p(z), Acta Arithmetica, Volume 62 (1992), pp. 61-71 | MR | Zbl

[14] G. Pólya; G. Szegö Problems and theorems in analysis I, Springer-Verlag, Berlin, 1972 | MR | Zbl

[15] P. Ribenboim Density results on families of diophantine equations with finitely many solutions, L’Enseignement Mathématique, Volume 39 (1993), pp. 3-23 | MR | Zbl

[16] T. Šalát Remarks on Steinhaus Property and Ratio Sets of Positive Integers, Czech. Math. J., Volume 50 (2000), pp. 175-183 | DOI | MR | Zbl

[17] T. Šalát; V. Toma Olivier’s theorem and statistical convergence, Annales Math. Blaise Pascal, Volume 10 (2003), pp. 305-313 | DOI | Numdam | MR | Zbl

[18] J. Michael Steele Probability theory and combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997 | MR | Zbl

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