Comptes Rendus
Analyse mathématique
Ensembles à grande intersection et ubiquité
[Sets with large intersection and ubiquity]
Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 447-452.

We define new classes of sets with large intersection, which generalize those introduced by K. Falconer. These classes contain the sets which are defined using homogeneous and heterogeneous ubiquitous systems. Such sets play an important role in Diophantine approximation and in multifractal analysis.

Nous définissons de nouvelles classes d'ensembles à grande intersection, qui généralisent celles introduites par K. Falconer. Ces classes contiennent les ensembles qui sont définis à partir de systèmes d'ubiquité homogènes et hétérogènes. De tels ensembles jouent un rôle important en approximation diophantienne et en analyse multifractale.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.09.002
Arnaud Durand 1

1 Laboratoire d'analyse et de mathématiques appliquées, université Paris XII, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France
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     title = {Ensembles \`a grande intersection et ubiquit\'e},
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Arnaud Durand. Ensembles à grande intersection et ubiquité. Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 447-452. doi : 10.1016/j.crma.2006.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.002/

[1] J.-M. Aubry; S. Jaffard Random wavelet series, Comm. Math. Phys., Volume 227 (2002), pp. 483-514

[2] J. Barral, S. Seuret, Heterogeneous ubiquitous systems in Rd and Hausdorff dimension, prépublication, 2004

[3] V.V. Beresnevich, S.L. Velani, A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, à paraître dans Ann. Math., 2005

[4] Y. Bugeaud Approximation by Algebraic Numbers, Cambridge University Press, 2004

[5] A. Durand, Sets with large intersection and ubiquity, prépublication, 2006

[6] A. Durand, Ubiquitous systems and metric number theory, prépublication, 2006

[7] K.J. Falconer Sets with large intersection properties, J. London Math. Soc., Volume 49 (1994) no. 2, pp. 267-280

[8] G. Harman Metric Number Theory, Clarendon Press, NY, 1998

[9] S. Jaffard The multifractal nature of Lévy processes, Probab. Theory Related Fields, Volume 114 (1999), pp. 207-227

[10] S. Jaffard On lacunary wavelet series, Ann. Appl. Probab., Volume 10 (2000) no. 1, pp. 313-329

[11] W.M. Schmidt Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc., Volume 110 (1964), pp. 493-518

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