Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T12:31:27.680Z Has data issue: false hasContentIssue false
  • English
  • Français

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: buczo@ludens.elte.hu
R. Daniel Mauldin
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203, U.S.A. E-mail: mauldin@unt.edu
Get access

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 every x εI∞, and Σ∞n = 1f(nx) >+∞ for almost every x εIf. 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

Information

Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 337 - 341
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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