Steinhaus theorem

In the mathematical field of real analysis, the Steinhaus theorem states that the difference set of a set of positive measure contains an open neighbourhood of zero. It was first proved by Hugo Steinhaus.

Statement

Let A be a Lebesgue-measurable set on the real line such that the Lebesgue measure of A is not zero. Then the difference set
contains an open neighbourhood of the origin.
The general version of the theorem, first proved by André Weil, states that if G is a locally compact group, and A ⊂ G a subset of positive Haar measure, then
contains an open neighbourhood of unity.
The theorem can also be extended to nonmeagre sets with the Baire property.

Corollary

A corollary of this theorem is that any measurable proper subgroup of is of measure zero.

Applications

A special case of the Steinhaus Theorem deals with the existence of arithmetic progressions in a set of positive Lebesgue measure. In particular, let, for some positive integer, be a set of positive Lebesgue measure. Then for any integer, contains a finite arithmetic progression of length.