Uniform distribution of generalized polynomials
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 \(q\).
scientific article; zbMATH DE number 483993