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 in R n {\displaystyle \mathbb {R} ^{n}} such that the Lebesgue measure of A is not zero. Then the difference set
A − A = { a − b ∣ a , b ∈ A } {\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, states that if G is a locally compact group, and A ⊂ G a subset of positive (left) Haar measure, then
A A − 1 = { a b − 1 ∣ a , b ∈ A } {\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 ( R , + ) {\displaystyle (\mathbb {R} ,+)} is of measure zero.
Applications A special case of the Steinhaus Theorem (and the Lebesgue density theorem) deals with the existence of arithmetic progressions in a set of positive Lebesgue measure. In particular, let E ⊂ R n {\displaystyle E\subset \mathbb {R} ^{n}} , for some positive integer n {\displaystyle n} , be a set of positive Lebesgue measure. Then for any integer N > 0 {\displaystyle N>0} , E {\displaystyle E} contains a finite arithmetic progression of length N + 1 {\displaystyle N+1} .
See also Falconer's conjecture
Notes
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