Steinhaus theorem

Short description: Mathematical theorem in real analysis

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

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

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.

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 $E\subset {\mathbb{ℝ}}^n$, for some positive integer $n$, be a set of positive Lebesgue measure. Then for any integer $N>0$, $E$ contains a finite arithmetic progression of length $N+1$.

Proof

Let $E\subset {\mathbb{ℝ}}^n$ be a set of positive Lebesgue measure, $\{a_1,a_2,\dots ,a_N\}$ be an arbitrary collection of unit vectors in ${\mathbb{ℝ}}^n$, and $ϵ\in (0,(2^N-1{)}^{-1})$. Also denote the $n$-dimensional Lebesgue measure by $m^n$. By inner regularity of the Lebesgue measure, we obtain a compact set $K_1\subset E$ such that $m^n(K_1)>0$, and by outer regularity an open set $U\supset K_1$ such that $m^n(U)\le (1+ϵ)m^n(K_1).$

Because $K_1$ is compact, the distance $R=d(K_1,U^c)$ is strictly positive. Let $\delta \in (0,R)$ be arbitrary, and consider the set $K_1+\delta a_1$. If this subset is not contained in $U$, then we would have

$$\begin{array}{r}d(K_1,U^c)<\Vert \delta a_1\Vert =\delta <R,\end{array}$$ which is a contradiction. Therefore, $K_1\cap (K_1+\delta a_1)\subset U$. This means that

$$\begin{array}{r}{m}^n(U)\ge m^n(K_1\cup (K_1+\delta a_1))=m^n(K_1)+m^n(K_1+\delta a_1)-m^n(K_1\cap (K_1+\delta a_1)).\end{array}$$

By translation invariance of the Lebesgue measure, we note that $m^n(K_1+\delta a_1)=m^n(K_1)$, and so

$$\begin{array}{r}{m}^n(K_1\cap (K_1+\delta a_1))\ge 2m^n(K_1)-m^n(U)\ge (1-ϵ)m^n(K_1).\end{array}$$

Since $ϵ<1$, we see that the measure on the left side is strictly positive, which means $K_1+\delta a_1\ne \mathrm{\varnothing }.$ Now for each $i=1,\dots ,N$, define the sets $K_{i+1}=K_i\cap (K_i+\delta a_i)$. By a generalization of the argument above, each $K_i$ is contained in $U$. Moreover, for each $i$, $m^n(K_i)\ge (1-(2^i-1))m^n(K_1)$ (a simple application of induction immediately yields this result) so that each $K_i$ is nonempty. This yields a nested sequence of sets $\mathrm{\varnothing }\ne K_{N+1}\subset K_N\subset \cdots \subset K_1\subset E$. Let $q\in K_{N+1}$. Since $K_{N+1}=K_N+\delta a_N$, $q-\delta a_N\in K_N$. Likewise, since $K_N=K_{N-1}+\delta a_{N-1}$, $q-\delta a_N-\delta a_{N-1}\in K_{N-1}$. Repeating this procedure iteratively and eventually denoting $p=q-\delta \sum _{i=1}^N{a}_i$, we recover the finite arithmetic progression $\{p,p+\delta a_1,p+\delta a_1+\delta a_2,\dots ,p+\delta a_1+\cdots +\delta a_N\}\subset E$ consisting of $N+1$ points. Hence, the proof concludes.

See also

• Falconer's conjecture

References

• "Sur les distances des points dans les ensembles de mesure positive" (in French). Fund. Math. 1: 93–104. 1920. doi:10.4064/fm-1-1-93-104. http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1111.pdf. .
• 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. 
• Sadhukhan, Arpan (2020). "An Alternative Proof of Steinhaus's Theorem". American Mathematical Monthly 127 (4): 330. doi:10.1080/00029890.2020.1711693. 

• 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