Weighted uniform densities
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 191-204.

We introduce the concept of uniform weighted density (upper and lower) of a subset A of * , 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) α–densities of A are obtained.

Nous introduisons la notion de densité uniforme pondé- rée (supérieure et inférieure) d’une partie A de N * , par rapport à une suite de poids (a n ). 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 α-densités (classiques) d’une partie A de N * .

Received:
Published online:
DOI: 10.5802/jtnb.581
Keywords: weighted uniform density, uniform density, weighted density, α–density.
Rita Giuliano Antonini 1; Georges Grekos 2

1 Università di Pisa Dipartimento di Matematica “L. Tonelli” Largo Bruno Pontecorvo 5 56127 Pisa, Italia
2 Université Jean Monnet 23, rue du Dr Paul Michelon 42023 St Etienne Cedex 2, France
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Rita Giuliano Antonini; Georges Grekos. Weighted uniform densities. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 191-204. doi : 10.5802/jtnb.581. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.581/

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