Comptes Rendus
Harmonic Analysis/Dynamical Systems
On systems of dilated functions
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1261-1263.

If f(x)=Zae2iπx satisfies ν1aν2Δ(ν)<, where Δ is the Erdös–Hooley function, we show that the series k=0ckf(kx) converges for almost every x, whenever the coefficient sequence verifies the condition

r(j=2r+12r+1cj2d(j)(logj)2)1/2<,
d being the divisor function. This strongly improves earlier related results.

Pour toute fonction f(x)=Zae2iπx telle que ν1aν2Δ(ν)<, où Δ est la fonction de Erdös–Hooley, nous montrons que la série k=0ckf(kx) converge presque partout dès que la suite des coefficients vérifie

r(j=2r+12r+1cj2d(j)(logj)2)1/2<,
d(n) désignant la fonction des diviseurs de n. Ceci améliore considérablement un certain nombre de résultats partiels précédemment obtenus.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.11.003

Michel J.G. Weber 1

1 IRMA, 7, rue René-Descartes, 67084 Strasbourg cedex, France
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Michel J.G. Weber. On systems of dilated functions. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1261-1263. doi : 10.1016/j.crma.2011.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.003/

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[2] I. Berkes; M. Weber On the convergence of ckf(nkx), Memoirs of the A.M.S., Volume 201 (2009) no. 943 (vi+72p)

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[7] M. Weber Dynamical Systems and Processes, IRMA Lectures in Mathematics and Theoretical Physics, vol. 14, European Mathematical Society Publishing House, 2009 (xiii+759p)

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