We introduce the concept of uniform weighted density (upper and lower) of a subset of , with respect to a given sequence of weights . 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) –densities of are obtained.
Nous introduisons la notion de densité uniforme pondé- rée (supérieure et inférieure) d’une partie de par rapport à une suite de poids Ce concept généralise la notion classique de la densité uniforme (pour laquelle les poids sont tous égaux à 1). Nous démontrons un théorème de comparaison de deux densités uniformes (ayant des suites de poids différentes) et un théorème de comparaison d’une densité pondérée uniforme et d’une densité pondérée classique (asymptotique ; non uniforme). Comme conséquence, nous obtenons un nouveau majorant et un nouveau minorant pour l’ensemble des densités (classiques) d’une partie de
@article{JTNB_2007__19_1_191_0,
author = {Giuliano Antonini, Rita and Grekos, Georges},
title = {Weighted uniform densities},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {191--204},
publisher = {Universit\'e Bordeaux 1},
volume = {19},
number = {1},
year = {2007},
doi = {10.5802/jtnb.581},
zbl = {1128.11005},
mrnumber = {2332061},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jtnb.581/}
}
TY - JOUR AU - Giuliano Antonini, Rita AU - Grekos, Georges TI - Weighted uniform densities JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 191 EP - 204 VL - 19 IS - 1 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.581/ DO - 10.5802/jtnb.581 LA - en ID - JTNB_2007__19_1_191_0 ER -
%0 Journal Article %A Giuliano Antonini, Rita %A Grekos, Georges %T Weighted uniform densities %J Journal de théorie des nombres de Bordeaux %D 2007 %P 191-204 %V 19 %N 1 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.581/ %R 10.5802/jtnb.581 %G en %F JTNB_2007__19_1_191_0
Giuliano Antonini, Rita; Grekos, Georges. Weighted uniform densities. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 191-204. doi : 10.5802/jtnb.581. http://archive.numdam.org/articles/10.5802/jtnb.581/
[1] R. Alexander, Density and multiplicative structure of sets of integers. Acta Arithm. 12 (1976), 321–332. | MR | Zbl
[2] T. C. Brown - A. R. Freedman, Arithmetic progressions in lacunary sets. Rocky Mountain J. Math. 17 (1987), 587–596. | MR | Zbl
[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
[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 | Zbl
[6] P. Ribenboim, Density results on families of diophantine equations with finitely many solutions. L’Enseignement Mathématique 39, (1993), 3–23. | Zbl
[7] H. Rohrbach - B. Volkmann, Verallgemeinerte asymptotische Dichten. J. Reine Angew. Math. 194 (1955), 195 –209. | MR | Zbl
[8] T. Šalát - V. Toma, A classical Olivier’s theorem and statistical convergence. Annales Math. Blaise Pascal 10 (2003), 305–313. | Numdam | Zbl
Cited by Sources:
