7 - Besicovitch and Kakeya sets

Summary

Introduction

The Kakeya problem has an interesting history. In 1917 Besicovitch was working on problems on Riemann integration, and was confronted with the following question: if f is a Riemann integrable function defined on the plane, is it always possible to find a pair of orthogonal coordinate axes with respect to which ∫ f(x,y)dx exists as a Riemann integral for all y, and with the resulting function of y also Riemann integrable? Besicovitch noticed that if he could construct a compact set F of plane Lebesgue measure zero containing a line segment in every direction, this would lead to a counter-example: For assume, by translating F if necessary, that F contains no segment parallel to and rational distance from either of a fixed pair of axes. Let f be the characteristic function of the set F₀ consisting of those points of F with at least one rational coordinate. As F contains a segment in every direction on which both F₀ and its complement are dense, there is a segment in each direction on which f is not Riemann integrable. On the other hand, the set of points of discontinuity of F is of plane measure zero, so f is Riemann integrable over the plane by the wellknown criterion of Lebesgue.

Besicovitch (1919) succeeded in constructing a set, known as a ‘Besicovitch set’, with the required properties. Owing to the unstable situation in Russia at the time, his paper received limited circulation, and the construction was later republished in Mathematische Zeitschrift (1928).