A note on uniform or Banach density. (English) Zbl 1239.11012

Let \({\mathbb N}\) be the set of positive integers, \(A\subset{\mathbb N}\) and \(I=[s,t]\subset{\mathbb N}\) an interval of length \(|I|=t-s\) and \(A(s,t)=A\cap[s,t]\). As a measure of a subset \(A\subset{\mathbb N}\) various density concepts are used, among them the Banach and the uniform density. The notions of the upper Banach density is defined in the paper as \[ \overline{b}(A)=\sup\{x\in[0,1];\;\forall\ell\in{\mathbb N}\;\exists I\subset{\mathbb N}: |I|\geq \ell \wedge |A\cap I|/|I|\geq x\}. \] The notion of the upper uniform density can be found in the literature in two forms, either as \[ \overline{a}(A)=\lim_{s\to\infty}(\limsup_{n\to\infty} A(n+1,n+s))/s \] or as \[ \overline{c}(A)=\lim_{s\to\infty}(\sup_{n\to\infty} A(n+1,n+s))/s. \] The aim of the paper is to show that all three values coincide. Dual results for the lower densities are also proved.


11B05 Density, gaps, topology
Full Text: DOI Numdam EuDML


[1] Bergelson, Vitaly, Sets of recurrence of \(\textbf{Z}^m\)-actions and properties of sets of differences in \(\textbf{Z}^m,\) J. London Math. Soc. (2), 31, 2, 295-304, (1985) · Zbl 0579.10029
[2] Bergelson, Vitaly, Ergodic Ramsey theory, Contemporary Math., 65, 63-87, (1987) · Zbl 0642.10052
[3] Bergelson, Vitaly; Host, B.; Kra, B., Multiple recurrence and nilsequences. with an appendix by imre ruzsa, Invent. Math., 160, 261-303, (2005) · Zbl 1087.28007
[4] Brown, T. C.; Freedman, A. R., Arithmetic progressions in lacunary sets, Rocky Mountain J. Math., 17, 587-596, (1987) · 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, XII, 1-6, (1990) · 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., 13, 374-382, (1951) · Zbl 0044.06003
[7] Dressler, R. E.; Pigno, L., Small sum sets and the Faber gap condition, Acta Sci. Math. (Szeged), 47, 233-237, (1984) · Zbl 0564.43006
[8] Fekete, M., Über die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten., Math. Zeitschr., 17, 228-249, (1923) · JFM 49.0047.01
[9] Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory, (1981), Princeton University Press, Princeton, N.J. · Zbl 0459.28023
[10] Gáliková, Z.; László, B.; Šalát, T., Remarks on uniform density of sets of integers, Acta Acad. Paed. Agriensis, Sectio Math., 23, 3-13, (2002) · Zbl 1012.11012
[11] Hegyvári, N., Note on difference sets in \(\mathbb{Z}^n,\) Periodica Math. Hungarica, 44, 183-185, (2002) · Zbl 1006.11006
[12] Jin, R., Nonstandard methods for upper Banach density problems, Journal of Number Theory, 91, 20-38, (2001) · Zbl 1071.11503
[13] Nair, R., On certain solutions of the Diophantine equation \(x - y = p(z),\) Acta Arithmetica, 62, 61-71, (1992) · Zbl 0776.11006
[14] Pólya, G.; Szegö, G., Problems and theorems in analysis I, (1972), Springer-Verlag, Berlin · Zbl 0236.00003
[15] Ribenboim, P., Density results on families of Diophantine equations with finitely many solutions, L’Enseignement Mathématique, 39, 3-23, (1993) · Zbl 0804.11026
[16] Šalát, T., Remarks on Steinhaus property and ratio sets of positive integers, Czech. Math. J., 50, 175-183, (2000) · Zbl 1034.11010
[17] Šalát, T.; Toma, V., Olivier’s theorem and statistical convergence, Annales Math. Blaise Pascal, 10, 305-313, (2003) · Zbl 1061.40001
[18] Steele, J. Michael, Probability theory and combinatorial optimization, 69, (1997), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA · Zbl 0916.90233
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.