Equidistributed Sequences

Abstract

In 1935, J.G. van der Corput asked a question about how to evenly distribute sequences of points. By 1945, Van Aardenne-Ehrenfest found an answer, which encouraged more mathematical research. The problem they explored was how to place points evenly on a circle and keep this even distribution when adding new points. In other words, van der Corput wondered if points could be placed on a line segment in such a way that any two subintervals of the same length would have almost the same number of points. Van Aardenne showed that perfect equal distribution is impossible, leading to more studies on the irregularities in point distribution in different areas of mathematics.

A lot of mathematical work has been done on studying sequences of points in the interval [0, 1) that are evenly distributed. This report looks at the idea that a sequence is evenly distributed if its first n points divide a circle into intervals that are roughly equal in length. The sequence X_k = log_2(2k − 1) mod 1 (section 2.2) was introduced by De Bruijn and Erdös in this context. We will show that the way the gaps between points in this sequence are structured is uniquely optimal in a certain way.

This thesis aims to uncover new insights through a mix of theoretical and numerical studies. The first sections focus on the work of Nicolaas Govert de Bruijn and Paul Erdös, especially their 1949 pa- per "Sequences of Points on a Circle". Later sections will present numerical findings and compare different methods for achieving the best way to split a circle evenly.

In short, this thesis aims to reproduce mathematical theories about evenly distributed sequences and the best ways to divide intervals. By merging these theories with new numerical techniques, it hopes to provide a new viewpoint on a classic topic, inspiring more investigation and study in this fascinating area of mathematics.