Acta Mathematica Hungarica
Volume/Issue:
Volume 117: Issue 1-2
Universally L1 good sequences with gaps tending to infinity
Author:
Z. Buczolich Eötvös Loránd University Department of Analysis Pázmány Péter Sétány 1/c 1117 Budapest Hungary

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Pages:
91–140
Online Publication Date:
11 Jul 2007
Publication Date:
01 Oct 2007
Article Category:
Research Article
DOI:
https://doi.org/10.1007/s10474-007-6068-8
Keywords:
ergodic theorem; universally good sequence; Banach density; 37A05; 28D05; 47A35
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Abstract  

We construct a sequence (nk) such that nk + 1 − nk → ∞ and for any ergodic dynamical system (X, Σ, �, T) and f ε L1(�) the averages
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lim _{N \to \infty } (1/N)\sum\nolimits_{k = 1}^N {f(T^{n_k } x)}$$ \end{document}
converge to ∝Xfd� for � almost every x. Since the above sequence is of zero Banach density this disproves a conjecture of J. Rosenblatt and M. Wierdl about the nonexistence of such sequences.
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Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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