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Steinhaus 1920 sur les distances des points dans
Steinhaus, Hugo, Sur les distances des points dans les ensembles de mesure positive. Fund. Math. 1 (1920), 93-104. DOI: 10.4064/fm-1-1-93-104. The scan prints no copyright or license line; the publisher's record offers the PDF under the link "Free download under CC-BY license" and names no Creative Commons version or URL (https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/1/1/92382/sur-les-distances-des-points-dans-les-ensembles-de-mesure-positive, read 2026-10-02), so the term is the Creative Commons Attribution license with its version unstated; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.
The held PDF is a scan of the twelve printed pages 93-104, watermarked "ICM Biblioteka Wirtualna Matematyki", with no text layer; PDF page k is printed page 92+k. Read status: claims checked for Theoremes I, II, III, VI, VII and VIII and the corollary on p. 100 on the page images (pp. 94-97, 99-100); the proofs of Theoremes I and VIII were read.
This French-language paper starts from Sierpinski's theorem that two linear sets of positive Lebesgue measure contain points, one from each, at rational distance, and develops generalizations. Theoreme I (p. 94) proves that every linear set of positive measure contains two distinct points at rational distance; the proof is a translation argument, since otherwise the translates E+1, E+1/2, ..., E+1/k would be pairwise disjoint and a bounded positive-measure part would give a set of infinite measure inside a bounded region, a contradiction. Section 1 extends this: Theoreme II (p. 95) finds k distinct points with pairwise rational distances for every k, and Theoreme III (p. 96) an infinite sequence of distinct points all of whose mutual distances are rational. Section 2 introduces the distance set of two sets (the distances between a point of one and a point of the other) and of a single set. Its Theoreme VI (p. 97) generalizes Sierpinski's theorem: for two sets A, B of positive measure and any everywhere-dense set C of numbers, some a in A and b in B have distance in C. From it the paper deduces in Theoreme VII (p. 99) that the distance set of two sets of positive measure contains an interval and in Theoreme VIII (p. 99) the result now known as the Steinhaus theorem: the distance set of a set of positive measure contains an interval whose left endpoint is 0, so its difference set contains a neighbourhood of zero. A corollary on p. 100 allows positive inner measure in place of positive measure in Theoremes VII and VIII. For Erdos problem 120, which asks whether for every infinite set A of reals some set of positive measure contains no affine copy of A, Theoreme VIII is the classical source for the fact that positive measure forces the difference set to contain an interval around zero, so every set of positive measure contains pairs at every small distance. The paper does not treat affine copies of finite sets of three or more points.
Bears on. #120
Results to transcribe.
- Theoreme I (p. 94): Every linear set of positive measure contains two distinct points a, b whose distance is rational; proved by a translation and measure-additivity contradiction.
- Theoremes II and III (pp. 95-96): A linear set of positive measure contains, for each natural number k, k distinct points with pairwise rational distances, and an infinite sequence of distinct points all of whose mutual distances are rational.
- Theoremes VII and VIII (p. 99): The distance set of two sets of positive measure contains an interval; the distance set of one set of positive measure contains an interval with left endpoint 0 (the Steinhaus theorem).