A note on uniform or Banach density
Grekos, Georges ; Toma, Vladimír ; Tomanová, Jana
Annales mathématiques Blaise Pascal, Tome 17 (2010), p. 153-163 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/ambp.280
Classification:  11B05 
@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},
     volume = {17},
     year = {2010},
     pages = {153-163},
     doi = {10.5802/ambp.280},
     mrnumber = {2674656},
     zbl = {1239.11012},
     language = {en},
     url = {http://dml.mathdoc.fr/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) pp. 153-163. doi : 10.5802/ambp.280. http://gdmltest.u-ga.fr/item/AMBP_2010__17_1_153_0/

[1] Bergelson, Vitaly Sets of recurrence of Z m -actions and properties of sets of differences in Z m , J. London Math. Soc. (2), Tome 31 (1985) no. 2, pp. 295-304 | Article | MR 809951 | Zbl 0579.10029

[2] Bergelson, Vitaly Ergodic Ramsey theory, Contemporary Math., Tome 65 (1987), pp. 63-87 | MR 891243 | Zbl 0642.10052

[3] Bergelson, Vitaly; Host, B.; Kra, B. Multiple recurrence and nilsequences. With an appendix by Imre Ruzsa, Invent. Math., Tome 160 (2005), pp. 261-303 | Article | MR 2138068 | Zbl 1087.28007

[4] Brown, T. C.; Freedman, A. R. Arithmetic Progressions in Lacunary Sets, Rocky Mountain J. Math., Tome 17 (1987), pp. 587-596 | Article | MR 908265 | Zbl 0632.10052

[5] Brown, T. C.; Freedman, A. R. The Uniform Density of Sets of Integers and Fermat’s Last Theorem, C. R. Math. Rep. Acad. Sci. Canada, Tome XII (1990), pp. 1-6 | MR 1043085 | Zbl 0701.11011

[6] De Bruijn, N. G.; Erdős, P. Some linear and some quadratic recursion formulas, I. Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math., Tome 13 (1951), pp. 374-382 | MR 47161 | Zbl 0044.06003

[7] Dressler, R. E.; Pigno, L. Small sum sets and the Faber gap condition, Acta Sci. Math. (Szeged), Tome 47 (1984), pp. 233-237 | MR 755578 | Zbl 0564.43006

[8] Fekete, M. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten., Math. Zeitschr., Tome 17 (1923), pp. 228-249 | Article | MR 1544613

[9] Furstenberg, H. 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] Gáliková, Z.; László, B.; Šalát, T. Remarks on uniform density of sets of integers, Acta Acad. Paed. Agriensis, Sectio Math., Tome 23 (2002), pp. 3-13 | MR 1956574 | Zbl 1012.11012

[11] Hegyvári, N. Note on difference sets in ℤ n , Periodica Math. Hungarica, Tome 44 (2002), pp. 183-185 | Article | MR 1918685 | Zbl 1006.11006

[12] Jin, R. Nonstandard methods for upper Banach density problems, Journal of Number Theory, Tome 91 (2001), pp. 20-38 | Article | MR 1869316 | Zbl 1071.11503

[13] Nair, R. On certain solutions of the diophantine equation x-y=p(z), Acta Arithmetica, Tome 62 (1992), pp. 61-71 | MR 1179010 | Zbl 0776.11006

[14] Pólya, G.; Szegö, G. Problems and theorems in analysis I, Springer-Verlag, Berlin (1972) | MR 1492447 | Zbl 0236.00003

[15] Ribenboim, P. Density results on families of diophantine equations with finitely many solutions, L’Enseignement Mathématique, Tome 39 (1993), pp. 3-23 | MR 1225254 | Zbl 0804.11026

[16] Šalát, T. Remarks on Steinhaus Property and Ratio Sets of Positive Integers, Czech. Math. J., Tome 50 (2000), pp. 175-183 | Article | MR 1745470 | Zbl 1034.11010

[17] Šalát, T.; Toma, V. Olivier’s theorem and statistical convergence, Annales Math. Blaise Pascal, Tome 10 (2003), pp. 305-313 | Article | Numdam | MR 2031274 | Zbl 1061.40001

[18] Steele, J. Michael Probability theory and combinatorial optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, CBMS-NSF Regional Conference Series in Applied Mathematics, Tome 69 (1997) | MR 1422018 | Zbl 0916.90233