Steinhaus theorem

The Steinhaus theorem states that if a measurable subset of Euclidean space has positive Lebesgue measure, its difference set contains a neighborhood of zero.

Difference set: For a set in a group, its difference set consists of all differences of pairs of its elements. The theorem's conclusion says this set contains every vector sufficiently close to zero.
Measurable set: A measurable set is a set to which a specified measure can consistently assign a size. The theorem requires measurability so its set has a well-defined Lebesgue measure.
Hugo Steinhaus: Hugo Steinhaus was a Polish mathematician who worked in analysis, probability, and related fields. The theorem bears his name and belongs to the measure-theoretic analysis he helped develop.
Additive combinatorics: Additive combinatorics studies additive structure in sets, especially through sums and differences. The theorem guarantees local additive structure in every positive-measure set.
Convolution: Convolution combines two functions by integrating products of one function with translates of another. Convolving an indicator with its reflection measures overlap between a set and its translates.
Positive measure: A set has positive measure when its measure is strictly greater than zero. This is the theorem's nontrivial size assumption.
Steinhaus theorem for topological groups: A generalization states that positive-measure subsets of suitable locally compact groups have difference sets containing a neighborhood of the identity. It extends the Euclidean conclusion from vectors to group operations.
Sumset: The sumset of two sets consists of all sums formed from one element of each set. Difference sets are sumsets formed by adding a set to the negation of itself.
Translation continuity in L¹: For an integrable function, translating its argument by a vector approaching zero makes it converge in the L¹ norm. It ensures the overlap function remains positive for all sufficiently small translations.
Neighborhood of zero: A neighborhood of zero is a set containing an open set around the origin. The theorem guarantees that every sufficiently small vector is a difference of two set elements.

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Lebesgue density theoremCompared with: It is another consequence of positive measure, but concerns sums and differences rather than local density.