×

zbMATH — the first resource for mathematics

On distribution of sequences of integers. (English) Zbl 0803.11038
Let \(\Gamma= \{h(j,m)\): \(0\leq j<m\), \(m\in \mathbb{N}\}\) be a system of non- negative real numbers. A sequence \(\omega= (w_ n)\) of positive integers is called \(\Gamma\)-distributed if and only if \[ \lim_{N\to\infty} {{\#\{n\leq N:\;w_ n\equiv j\pmod m\}} \over N} = h(j,m), \] for every \(j,m\in \mathbb{N}\) and \(0\leq j<m\). Applying a general metric result for uniform distribution in compact spaces, the authors show the existence of a \(\Gamma\)-distributed sequence \(\omega\) provided that \(\omega\) satisfies \(h(0,1) =1\) and \(h(j,m)= \sum_{r=0}^{k-1} h(j+ rm, km)\) for every \(k,m\in \mathbb{N}\), \(0\leq j<m\). The authors make use of an interesting connection to Novoselov’s theory of polyadic numbers. In a final chapter transformations of \(\Gamma\)-distributed sequences are studied.
Reviewer: R.F.Tichy (Graz)
MSC:
11K06 General theory of distribution modulo \(1\)
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] BUCK R. C.: The measure theoretical approach to the density. Amer. J. Math. LXVIII (1946), 560-580. · Zbl 0061.07503
[2] JACOBS K.: Measure and Integral. Academic Press, New York-San Francisco-London, 1987.
[3] KUIPERS L.-NIEDERREITER H.: Uniform Distribution of Sequences. J.Wiley & Sons, New York-London-Sydney-Toronto, 1974. · Zbl 0281.10001
[4] MEIJER H. G.: The discrepancy of g-adic sequence. Indag. Math. 30 (1968), 54-66. · Zbl 0155.37801
[5] MEIJER H. G.: Uniform distribution of g-adic numbers. Indag. Math. 29 (1967), 535-546. · Zbl 0207.05802
[6] NIVEN I.: Uniform distribution of sequences of integers. Trans. Amer. Math. Soc. 98 (1961), 52-61. · Zbl 0096.03102
[7] NOVOSELOV E. V.: Topological theory of divisibility. (Russian), Uchen. Zap. Elabuz. PI 8 (1960), 3-23.
[8] PAŠTÉKA M., ŠALÁT T.: Buck’s measure density and sets of positive integers containing arithmetic progression. Math. Slovaca 41 (1991), 283-293. · Zbl 0761.11004
[9] POSTNIKOV A. G.: Introduction to Analytic Number Theory. ( · Zbl 0641.10001
[10] PRUFER H.: Neue Begrüdung der algebraischen Zahlentheorie. Math. Ann. 94 (1925), 198-243. · JFM 51.0140.14
[11] SCHOENBERG, I: Über die asymptotische Verteilung reeler Zahlen mod 1. Math. Z. 28 (1928), 171-199. · JFM 54.0212.02
[12] WEYL H.: Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352. · JFM 46.0278.06
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.