Uniform distribution of generalized polynomials
type: Item
instance of: scholarly article
title: Uniform distribution of generalized polynomials
zbMATH Open document ID: 0797.11064
author: Inger Johanne Håland
published in: Journal of Number Theory
publication date: 1994-10-20T00:00:00Z
review text: The background of this paper is a classical result of H. Weyl in the theory of uniform distribution of sequences modulo one: if \(p\) is a polynomial over \(\mathbb{R}\) with \(\deg p \geq 1\), with at least one irrational coefficient, then the sequence \((p(n))_{n \geq 0}\) is uniformly distributed modulo one. The author considers the uniform distribution of sequences \((q(n))_{n \geq 0}\), where \(q\) is a generalized polynomial. This class of functions arises from real-valued polynomials by adding the operation of taking the greatest integer to the arithmetical operations. Typical examples are \(q(x) = [p(x)] \beta\), or \(q(x) = [ax] [bx]c\). Due to \textit{W. Veech} [Trans. Am. Math. Soc. 161, 63-70 (1971; Zbl 0229.10019)] we know necessary and sufficient conditions for \(([p(n)]\beta)_{n\geq 0}\) to be well-distributed. In this paper, the author applies van der Corput's difference theorem [see \textit{L. Kuipers} and \textit{H. Niederreiter}, Uniform Distribution of Sequences, (1974; Zbl 0281.10001)] to study the general case. The symbolic representation of a sequence \((q(n))_{n \geq 0}\), \(q\) a generalized polynomial, is not unique. This fact causes certain technical and notational difficulties that cannot be discussed in this short review. As an example of the numerous results, for simple generalized polynomials \(q\) of degree \(2\) \((q(n) = \gamma n^ 2\), \(q(n) = [\alpha n] \gamma n\), \(q(n) = [\alpha n] [\beta n] \gamma)\), the author has been able to give necessary and sufficient conditions for the uniform distribution modulo one of \((q(n))_{n \geq 0}\) in terms of the coefficients of \(q\).
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 483993
zbMATH Keywords: uniform distribution of sequences
zbMATH Keywords: generalized polynomial
zbMATH Keywords: van der Corput's difference theorem
zbMATH Keywords: uniform distribution modulo one
reviewed by: Peter Hellekalek
MaRDI profile type: Publication
OpenAlex ID: W2158593242
DOI: 10.1006/JNTH.1993.1082
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description: scientific article; zbMATH DE number 483993