Generalizations of Steinhaus' theorem

Kevin A. Smith

In 1920 the Polish mathematician Hugo Steinhaus (1887-1972) proved that the distance set of a subset of the real line with positive Lebesgue measure has non-empty interior. In this thesis we examine results which generalize the original Steinhaus' theorem; in particular, we attempt to find a structure in which both measure-theoretic extensions of Steinhaus' theorem and their corresponding category-theoretic results can be expressed. Also we examine perfect sets, which appear to provide a convenient basis from which to attempt further generalizations of Steinhaus' theorem. The thesis is divided into four chapters. The first chapter contains an introduction to the thesis, together with a listing of general references and frequently used concepts. In the second chapter we give some of the general properties of sum and distance sets. The main chapter of the thesis is the third. In it we investigate an abstraction of the classical notion of Baire category which still retains the basic results such as the Banach category theorem. Next, we describe recent measure-theoretic extensions of Steinhaus' theorem. Lastly, we consider generalizations of the category version of Steinhaus' theorem. due to Sophie Piccard, which states that the distance set of a subset of second category Baire of the real line has a non-empty interior. In the final chapter we examine perfect sets. Here we first show how perfect sets provide a convenient setting for generalizing Steinhaus' theorem. Then we examine various representations of perfect sets on the real line.

Generalizations of Steinhaus' theorem

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