On the development of the theory of uniform distribution during the years 1909 to 1916
type: Item
instance of: scholarly article
title: On the development of the theory of uniform distribution during the years 1909 to 1916
zbMATH Open document ID: 0606.10001
DOI: 10.1007/BF00412261
author: Christa Binder
author: Edmund Hlawka
published in: Archive for History of Exact Sciences
publication date: 1986-01-01T00:00:00Z
review text: A sequence \((x_ n)\) is uniformly distributed (equidistributed) in a real interval \(I\) if, for each subinterval \(J\), the limiting frequency of its appearance in \(J\) is proportional to the length of \(J\). In analytic terms, the condition can be reformulated as \[ \lim_{N\to \infty}\sum^N_{n=1} f(x_n)=\int_{I}f(x)\,dx. \] Such sequences occur in a variety of contexts from perturbations of planetary systems (Lagrange) and heat diffusion to geometric and analytic number theory. This paper presents a particularly thorough study of the early history which reveals how the strands are woven together. Part I treats the work of P. Bohl (1865--1921, Latvian), H. Weyl, F. Bernstein, G. H. Hardy and J. E. Littlewood. Part II turns its focus on the major Mathematische Annalen paper of H. Weyl. Part III discusses other developments due to Hardy and Littlewood (Gaussian sums) and H. Bohr. The paper concludes with nine brief biographical notes and an appendix reprinting two short papers of W. Sierpiński (1909, 1910) and one of Hardy and Littlewood (1916).
Mathematics Subject Classification ID: 11-03
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 3979139
zbMATH Keywords: uniformly distributed sequences
zbMATH Keywords: P. Bohl
zbMATH Keywords: H. Weyl
zbMATH Keywords: F. Bernstein
zbMATH Keywords: G. H. Hardy
zbMATH Keywords: J. E. Littlewood
zbMATH Keywords: W. Sierpinski
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
cites work: John Edensor Littlewood
cites work: Q5549011
cites work: Mean Motion
cites work: Mean Motion. II
cites work: Ramifications, old and new, of the eigenvalue problem
cites work: Elementary Note on Prime Number Problems of Vinogradoff's Type
description: scientific article; zbMATH DE number 3979139