A recurrence property of smooth functions
type: Item
instance of: scholarly article
title: A recurrence property of smooth functions
zbMATH Open document ID: 0653.10036
DOI: 10.1007/BF02767351
author: Daniel Berend
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Using a result by \textit{R. C. Baker} [Journées Arithmétiques, Exeter 1980, Lond. Math. Soc. Lect. Note Ser. 56, 226--231 (1982; Zbl 0489.10028)] the author proves that if \(P\), \(Q\) are real polynomials of degrees \(d\) and \(e\) respectively and if \(f\) is a periodic real function which is \(s\) times differentiable at \(Q(0)\) with \(s\geq 7de^3 \log 14e^3\) then for any \(\varepsilon >0\) there exists an integer \(n\geq 1\) such that \[ \| P(n)f(Q(n)) - P(0)f(Q(0))\| < \varepsilon \] where \(\displaystyle\| \alpha \| =\min_{k\in\mathbb{Z}}| \alpha - k|\) for real \(\alpha\). The author claims that his result gives an affirmative answer to a question raised by \textit{H. Furstenberg} and \textit{B. Weiss} [Acta Arith. 49, No. 4, 413--426 (1988; Zbl 0653.10035), see the preceding review].
reviewed by: Ming-Chit Liu
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11J54
zbMATH DE Number: 4065096
zbMATH Keywords: recurrence properties of polynomials
zbMATH Keywords: small fractional parts of polynomials
zbMATH Keywords: Weyl's equidistribution theorem
zbMATH Keywords: IP-sets
MaRDI profile type: Publication
cites work: FRACTIONAL PARTS OF SEVERAL POLYNOMIALS
cites work: Q3950637
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cites work: Q3909268
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description: scientific article; zbMATH DE number 4065096