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

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

contains an open neighbourhood of the origin.

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

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

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

See also

• Falconer's conjecture

References

• Steinhaus, Hugo (1920). "Sur les distances des points dans les ensembles de mesure positive" (PDF). Fund. Math. (in French). 1: 93–104. doi:10.4064/fm-1-1-93-104..
• Weil, André (1940). L'intégration dans les groupes topologiques et ses applications. Hermann.
• Stromberg, K. (1972). "An Elementary Proof of Steinhaus's Theorem". Proceedings of the American Mathematical Society. 36 (1): 308. doi:10.2307/2039082. JSTOR 2039082.
• Sadhukhan, Arpan (2020). "An Alternative Proof of Steinhaus's Theorem". American Mathematical Monthly. 127 (4): 330. arXiv:1903.07139. doi:10.1080/00029890.2020.1711693. S2CID 84845966.

• Väth, Martin (2002). Integration theory: a second course. World Scientific. ISBN 981-238-115-5.
• Yueh-Shin, Lee,(1994). Counting Bipartite Steinhaus Graphs. National Chiao Tung University . https://hdl.handle.net/11296/afmq86