Weighted uniform densities

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 191-204
  • ISSN: 1246-7405

Abstract

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

How to cite

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Giuliano Antonini, Rita, and Grekos, Georges. "Weighted uniform densities." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 191-204. <http://eudml.org/doc/249952>.

@article{GiulianoAntonini2007,
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.},
affiliation = {Università di Pisa Dipartimento di Matematica “L. Tonelli” Largo Bruno Pontecorvo 5 56127 Pisa, Italia; Université Jean Monnet 23, rue du Dr Paul Michelon 42023 St Etienne Cedex 2, France},
author = {Giuliano Antonini, Rita, Grekos, Georges},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {weighted uniform density; uniform density; weighted density; $\alpha $–density},
language = {eng},
number = {1},
pages = {191-204},
publisher = {Université Bordeaux 1},
title = {Weighted uniform densities},
url = {http://eudml.org/doc/249952},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Giuliano Antonini, Rita
AU - Grekos, Georges
TI - Weighted uniform densities
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 191
EP - 204
AB - 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.
LA - eng
KW - weighted uniform density; uniform density; weighted density; $\alpha $–density
UR - http://eudml.org/doc/249952
ER -

References

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  2. T. C. Brown - A. R. Freedman, Arithmetic progressions in lacunary sets. Rocky Mountain J. Math. 17 (1987), 587–596. Zbl0632.10052MR908265
  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. Zbl0701.11011
  4. R. Giuliano Antonini - M. Paštéka, A comparison theorem for matrix limitation methods with applications. Uniform Distribution Theory 1 no. 1 (2006), 87–109. Zbl1146.11005
  5. C. T. Rajagopal, Some limit theorems. Amer. J. Math. 70 (1948), 157–166. Zbl0041.18301MR23930
  6. P. Ribenboim, Density results on families of diophantine equations with finitely many solutions. L’Enseignement Mathématique 39, (1993), 3–23. Zbl0804.11026
  7. H. Rohrbach - B. Volkmann, Verallgemeinerte asymptotische Dichten. J. Reine Angew. Math. 194 (1955), 195 –209. Zbl0064.28003MR70647
  8. T. Šalát - V. Toma, A classical Olivier’s theorem and statistical convergence. Annales Math. Blaise Pascal 10 (2003), 305–313. Zbl1061.40001

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