In this note we present and comment three equivalent definitions of the so called uniform or Banach density of a set of positive integers.
Classification : 11B05
Mots clés : Banach density, uniform density
@article{AMBP_2010__17_1_153_0, author = {Grekos, Georges and Toma, Vladim\'\i r and Tomanov\'a, Jana}, title = {A note on uniform or Banach density}, journal = {Annales Math\'ematiques Blaise Pascal}, pages = {153--163}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {17}, number = {1}, year = {2010}, doi = {10.5802/ambp.280}, zbl = {1239.11012}, mrnumber = {2674656}, language = {en}, url = {www.numdam.org/item/AMBP_2010__17_1_153_0/} }
Grekos, Georges; Toma, Vladimír; Tomanová, Jana. A note on uniform or Banach density. Annales Mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 153-163. doi : 10.5802/ambp.280. http://www.numdam.org/item/AMBP_2010__17_1_153_0/
[1] Sets of recurrence of -actions and properties of sets of differences in , J. London Math. Soc. (2), Volume 31 (1985) no. 2, pp. 295-304 | Article | MR 809951 | Zbl 0579.10029
[2] Ergodic Ramsey theory, Contemporary Math., Volume 65 (1987), pp. 63-87 | MR 891243 | Zbl 0642.10052
[3] Multiple recurrence and nilsequences. With an appendix by Imre Ruzsa, Invent. Math., Volume 160 (2005), pp. 261-303 | Article | MR 2138068 | Zbl 1087.28007
[4] Arithmetic Progressions in Lacunary Sets, Rocky Mountain J. Math., Volume 17 (1987), pp. 587-596 | Article | MR 908265 | Zbl 0632.10052
[5] 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 1043085 | Zbl 0701.11011
[6] Some linear and some quadratic recursion formulas, I. Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math., Volume 13 (1951), pp. 374-382 | MR 47161 | Zbl 0044.06003
[7] Small sum sets and the Faber gap condition, Acta Sci. Math. (Szeged), Volume 47 (1984), pp. 233-237 | MR 755578 | Zbl 0564.43006
[8] Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten., Math. Zeitschr., Volume 17 (1923), pp. 228-249 | Article | MR 1544613
[9] Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981 (M. B. Porter Lectures) | MR 603625 | Zbl 0459.28023
[10] Remarks on uniform density of sets of integers, Acta Acad. Paed. Agriensis, Sectio Math., Volume 23 (2002), pp. 3-13 | MR 1956574 | Zbl 1012.11012
[11] Note on difference sets in , Periodica Math. Hungarica, Volume 44 (2002), pp. 183-185 | Article | MR 1918685 | Zbl 1006.11006
[12] Nonstandard methods for upper Banach density problems, Journal of Number Theory, Volume 91 (2001), pp. 20-38 | Article | MR 1869316 | Zbl 1071.11503
[13] On certain solutions of the diophantine equation , Acta Arithmetica, Volume 62 (1992), pp. 61-71 | MR 1179010 | Zbl 0776.11006
[14] Problems and theorems in analysis I, Springer-Verlag, Berlin, 1972 | MR 1492447 | Zbl 0236.00003
[15] Density results on families of diophantine equations with finitely many solutions, L’Enseignement Mathématique, Volume 39 (1993), pp. 3-23 | MR 1225254 | Zbl 0804.11026
[16] Remarks on Steinhaus Property and Ratio Sets of Positive Integers, Czech. Math. J., Volume 50 (2000), pp. 175-183 | Article | MR 1745470 | Zbl 1034.11010
[17] Olivier’s theorem and statistical convergence, Annales Math. Blaise Pascal, Volume 10 (2003), pp. 305-313 | Article | Numdam | MR 2031274 | Zbl 1061.40001
[18] Probability theory and combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, Volume 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997 | MR 1422018 | Zbl 0916.90233