On the development of the theory of uniform distribution during the years 1909 to 1916
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).
scientific article; zbMATH DE number 3979139