Jackson 2002 sets meeting isometric copies lattice exactly one point

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theorem_1_1: Jackson and Mauldin's theorem, proved in ZFC, that some set S in the plane meets every isometric copy of the integer lattice Z^2 in exactly one point, so that S is a Steinhaus set.

theorem_1_2: Jackson and Mauldin's strengthening of their Theorem 1.1, proved in ZFC: some planar set meets every isometric copy of Z^2, and no two of its distinct points are at a distance whose square is an integer.

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Steve Jackson and R. Daniel Mauldin, Sets meeting isometric copies of the lattice $\mathbb Z^2$ in exactly one point, Proc. Natl. Acad. Sci. USA 99 (2002), no. 25, 15883--15887, DOI 10.1073/pnas.222551699.

The copy read for this card is the authors' TeX preprint of the paper, dated "August 22, 2002" in its footer and distilled from pnasshort3.dvi, eleven pages numbered 1--11. Its text layer is font-garbled, so the preprint was read on the page images; the published version was not compared, and the page numbers below are the preprint's. Provenance: downloaded in September 2026; the download URL was not recorded; 257,630 bytes. The preprint prints no copyright or license line (pp. 1 and 11 read on the page images); its download URL was not recorded, so no hosting page was consulted, and the publisher's page describes the published version, not the preprint; the term is unstated.

Read status. Claims checked: Theorems 1.1 and 1.2 and Lemma 1.3 were read clause by clause on the page images of pp. 1--2. The rest of the preprint, pp. 3--11, was read on the page images; the proofs were read but not checked step by step.

Result pages: Theorem 1.1 (p. 1) and Theorem 1.2 (p. 1).

Contents

Theorem 1.1 is the statement Problem 215 asks for.

• Introduction (p. 1): in the 1950s Steinhaus posed the problem "Is there a set $S$ in the plane such that every set congruent to $\mathbb Z^2$ has exactly one point in common with $S$?"; it "seems to have first appeared" in Sierpiński's 1958 paper [14], and it "has remained unsolved until now".
• Theorem 1.1 (ZFC) (p. 1): some $S\subseteq\mathbb R^2$ meets every isometric copy $L$ of the integer lattice $\mathbb Z^2$ in exactly one point, $|S\cap L|=1$. Such an $S$ is called a Steinhaus set; whether a Lebesgue measurable Steinhaus set exists "remains unsolved" (p. 1), and the paper cites Kolountzakis and Wolff [12] for the absence of a measurable Steinhaus set in the higher-dimensional version of the problem for the standard lattice.
• Theorem 1.2 (ZFC) (p. 1), stated as a strengthening of Theorem 1.1: there is $S\subseteq\mathbb R^2$ such that (1) $S\cap L\ne\emptyset$ for every isometric copy $L$ of $\mathbb Z^2$, and (2) for all distinct $z_1,z_2\in S$, $\rho(z_1,z_2)^2\notin\mathbb Z$, where $\rho$ is the Euclidean distance (a "lattice distance" is a number $\sqrt{n^2+m^2}$ with $n,m\in\mathbb Z$).
• Lemma 1.3 (A) (p. 2): the same two properties can be achieved for the countable family $\mathcal L_{\mathbb Q}$ of rational translates $\mathbb Z^2+(r,s)$, $r,s\in\mathbb Q$, by elementary number theory and combinatorics; the proof constructs a selector $f$ on $(\mathbb Q/\mathbb Z)^2$ with $\rho(f(x),f(y))^2\notin\mathbb Z$ for $x\ne y$ and reduces to the subgroup of $\mathbb Q\times\mathbb Q$ whose denominators are divisible only by primes congruent to $1$ modulo $4$.
• Closing (p. 11): the authors thank the referees for possible simplifications of the proof and for the simplified proof of Lemma A (Lemma 1.3) that the paper presents, and Robert M. Solovay for comments; the reference list names the detailed version, [9] Jackson and Mauldin, J. Amer. Math. Soc., "to appear".

Compiled scope

All eleven pages were read on the page images. Pages 2--10 carry the proof of Lemma 1.3 (pp. 2--4) and of Theorem 1.2 (ending on p. 10), through Lemmas 1.4, 1.7--1.10, Definition 1.6 and Claim 1.11. Lemma 1.5 (p. 6), which the paper says is stronger than the proof needs, the proofs of Lemma 1.9 and the details of Claim 1.11 are referred to the detailed paper [9], which was not read. The proofs were not checked step by step, nothing here is independently reviewed, and the published text was not compared with the preprint.

Bears on. #215: the problem asks for a planar set every translated and rotated copy of which contains exactly one point of $\mathbb Z^2$. Theorem 1.1, proved in ZFC, gives a set meeting every isometric copy of $\mathbb Z^2$ in exactly one point, which is equivalent to every isometric image of the set containing exactly one lattice point, and so gives a set of the kind the problem asks for (the Theorem 1.1 page writes out the equivalence). Theorem 1.2 gives such a set with no two points at a distance whose square is an integer. The paper leaves open whether a Lebesgue measurable such set exists.

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