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.^[1]

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

${\displaystyle A-A=\{a-b\mid a,b\in A\}}$

contains an open neighbourhood of the origin.

The general version of the theorem, first proved by André Weil,^[2] states that if G is a locally compact group, and A ⊂ G a subset of positive (left) Haar measure, then

${\displaystyle AA^{-1}=\{ab^{-1}\mid a,b\in A\}}$

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 ${\displaystyle (\mathbb {R} ,+)}$ is of measure zero.

See also

• Falconer's conjecture