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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.

MSC:

11B05 Density, gaps, topology
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[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
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