staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Rita Giuliano Antonini; Georges Grekos
Weighted uniform densities
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 191-204, doi: 10.5802/jtnb.581
Article PDF | Reviews MR 2332061 | Zbl 1128.11005
Keywords: weighted uniform density, uniform density, weighted density, $\alpha $–density.

Résumé - Abstract

We introduce the concept of uniform weighted density (upper and lower) of a subset $A$ of ${\mathbb{N}}^*$, with respect to a given sequence of weights $(a_n)$. This concept generalizes the classical notion of uniform density (for which the weights are all equal to 1). We also prove a theorem of comparison between two weighted densities (having different sequences of weights) and a theorem of comparison between a weighted uniform density and a weighted density in the classical sense. As a consequence, new bounds for the set of (classical) $\alpha $–densities of $A$ are obtained.

Bibliography

[1] R. Alexander, Density and multiplicative structure of sets of integers. Acta Arithm. 12 (1976), 321–332. Article |  MR 211979 |  Zbl 0189.04404
[2] T. C. Brown - A. R. Freedman, Arithmetic progressions in lacunary sets. Rocky Mountain J. Math. 17 (1987), 587–596.  MR 908265 |  Zbl 0632.10052
[3] 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 XII (1990), 1–6.  Zbl 0701.11011
[4] R. Giuliano Antonini - M. Paštéka, A comparison theorem for matrix limitation methods with applications. Uniform Distribution Theory 1 no. 1 (2006), 87–109.
[5] C. T. Rajagopal, Some limit theorems. Amer. J. Math. 70 (1948), 157–166.  MR 23930 |  Zbl 0041.18301
[6] P. Ribenboim, Density results on families of diophantine equations with finitely many solutions. L’Enseignement Mathématique 39, (1993), 3–23.  Zbl 0804.11026
[7] H. Rohrbach - B. Volkmann, Verallgemeinerte asymptotische Dichten. J. Reine Angew. Math. 194 (1955), 195 –209. Article |  MR 70647 |  Zbl 0064.28003
[8] T. Šalát - V. Toma, A classical Olivier’s theorem and statistical convergence. Annales Math. Blaise Pascal 10 (2003), 305–313. Cedram |  Zbl 1061.40001