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On the convergence of Σ∞n = 1f(nx) for measurable functions

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Zoltán Buczolich  and
R. Daniel Mauldin
Zoltán Buczolich
Affiliation:
Department of Analysis, Eőtvős Loránd University, Budapest, Hungary. E-mail: [email protected]
R. Daniel Mauldin
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203, U.S.A. E-mail: [email protected]
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Abstract

Questions of Haight and of Weizsäcker are answered in the following result. There exists a measurable function f: (0, + ∞) → {0,1} and two non-empty intervals IFI∞⊂[½,1) such that Σ∞n = 1f(nx) = +∞ for everyx εI∞, and Σ∞n = 1f(nx) >+∞ for almost every xεIf. The function f may be taken to be the characteristic function of an open set E.

MSC classification

Secondary: 28A35: Measures and integrals in product spaces
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 337 - 341
Copyright
Copyright © University College London 1999

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References

[BKM] Buczolich, Z., Kahane, J.-P. and Maudlin, R. D.. Sur les series de translates de fonctions positives. C.R. Acad. Sci. Paris Serie I, 329 (1999), 261–264.CrossRefGoogle Scholar
[C] Cassels, J. W. S.. An Introduction to Diophantine Approximation. (Cambridge University Press, 1957).Google Scholar
[F-H] Hyde, A. R. and Fine, N. J.. Solution of a problem proposed by K. L. Chung. Amer. Math. Monthly, 64 (1957), 119–120.Google Scholar
[HI] Haight, J. A.. A linear set of infinite measure with no two points having integral ratio. Mathematika, 17 (1970), 133–138.CrossRefGoogle Scholar
[H2] Haight, J. A.. A set of infinite measure whose ratio set does not contain a given sequence. Mathematika, 22 (1975), 195–201.CrossRefGoogle Scholar
[W] Weizsäcker, H. v.. Zum Konvergenzverhalten der Reihe für λ-messbare Funklionenf: f: ℝ+'→+. (Diplomarbeit, Universitat Miinchen, 1970.)Google Scholar