Uniform distribution of sequences of integers in residue classes (Q794688)

The theory of uniform distribution (u. d.) of sequences of integers was initiated by \textit{I. Niven} [Trans. Am. Math. Soc. 98, 52--61 (1961; Zbl 0096.03102)] who called a sequence of integers uniformly distributed (u. d.) modulo an integer \(m\geq 2\) if each residue class mod m contains asymptotically the same share of terms of the sequence. The first expository account of this theory was given in the book of \textit{L. Kuipers} and the reviewer [Uniform distribution of sequences (1974; Zbl 0281.10001)]. The related notion of weak u. d. mod m, in which only the distribution over the reduced residue classes mod m is considered, was introduced by the author [Acta Arith. 12, 269--279 (1967; Zbl 0147.29802)]. The present monograph contains an authoritative account of what is known about u. d. and weak u. d. of sequences of integers. After a review of basic properties in Chapter 1, sequences of polynomial values are studied in Chapter 2. Since for a polynomial \(f\in\mathbb{Z}[x]\) the sequence \(f(1),f(2),\ldots\) is u. d. mod \(m\) if and only if \(f\) permutes the residue classes mod \(m\), this leads to the study of polynomials possessing the latter property, i.e. of permutation polynomials mod \(m\). Some results about permutation polynomials that the author deems relevant in the given context are presented. Comprehensive treatments of the more algebraic aspects of permutation polynomials can be found in the books of \textit{H. Lausch} and \textit{W. Nöbauer} [Algebra of polynomials. Amsterdam etc.: North-Holland (1973; Zbl 0283.12101)] and \textit{R. Lidl} and the reviewer [Finite fields. Cambridge etc.: Cambridge University Press (1984; Zbl 0554.12010)]. Chapter 3 is devoted to u. d. mod \(m\) of sequences of integers satisfying linear recurrence relations, a subject that was initiated and studied extensively by L. Kuipers and his school. A detailed proof is given of the characterization of second-order linear recurring sequences that are u. d. mod \(m\). In Chapter 4 it is shown how E. Wirsing's mean-value theorem for multiplicative functions leads to a criterion of H. Delange for u. d. mod \(m\) of integer-valued additive functions. Chapters 5 and 6 treat weak u. d. mod \(m\) of multiplicative functions and contain mostly results of the author and his collaborators. A detailed bibliography and a subject index round off this monograph (note that the reference on p. 120 should be to L. J. Rogers rather than G. L. Rogers). The addenda on the last page list results of which the author only became aware after the completion of the original manuscript. It would have been more helpful for the reader if these items had been inserted as footnotes on the appropriate pages.