Two extensions of Steinhaus's theorem

Abstract

In 1920 H. Steinhaus [Sur les distances des points de mesure positive, Fundamenta Mathematicae 1 (1920) 93-104.] proved the following result: ”Let A be a Lebesgue measurable set of positive measure. Then there exist at least two points in A such that the distance between them is a rational number”. In this paper we shall prove that there exists a sequence (xn)n>1 of different points in A such that the distance between any two of them is a rational number. Further, we shall extend our result to the case when A is a set with the Baire property (non-necessarily Lebesgue measurable).