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Source. Theorem 1.1, p. 1, of Rachel Greenfeld, Marina Iliopoulou and Sarah Peluse, On integer distance sets, arXiv:2401.10821v3 (25 August 2025), the version named on the source card.

Read depth. Claims checked: the statement and the definition of an integer distance set were read clause by clause on the printed page, and the proof (pp. 30--33) was read for structure only. Nothing here is independently reviewed.

Statement

A set $S\subset\mathbb{R}^2$ is an integer distance set when the Euclidean distance between every pair of its points is an integer (p. 1).

Theorem 1.1 (Structure theorem, p. 1, quoted). "Let $S\subset[-N,N]^2$ be an integer distance set. Then, either $|S|=O\bigl((\log N)^{O(1)}\bigr)$, or else there exists a line or circle $C\subset\mathbb{R}^2$ such that $|S\setminus C|=O\bigl((\log\log N)^2\bigr)$."

Remark 1.2 (p. 1) says the result can be extended to subsets of the plane whose pairwise distances are rationals of height at most $N$, with the details left to the reader. The paper motivates the theorem (p. 1) by the observation that every integer distance set known so far has all but up to four of its points on one line or circle. Remark 2.4 (p. 10) says that no bound better than $|S\setminus C|\ll(\log N)^{O(1)}$ seems attainable by the paper's methods, since some polynomial dependence on the degree is necessary in the determinant-method bounds of Theorems 2.2 and 2.3.

Proof pointer

Pp. 30--33, following the outline of Section 2. Fix $k$ points of $S$ with $k\ll\log\log N$; after a linear change of variables, each point of $S$ together with its $k$ distances to those points becomes a rational point of height $\ll N^2$ on the closure of a surface $X_k$, which is irreducible of degree $2^k$ (Lemma 3.1, p. 10). The determinant method for surfaces (Theorem 2.2, p. 8) and a projection cover $S$ by $\ll e^{O(k)}N^{3/2^{k/2+1}}$ irreducible curves of comparable degree. For a curve that is not a line or circle, or is one with enough points of $S$ off it, a fresh choice of the $k$ points lifts it to an irreducible curve of degree at least $2^k$ (Lemmas 4.2 and 4.3, p. 17), on which the bound of Castryck, Cluckers, Dittmann and Nguyen for curves (Theorem 2.3, p. 9) leaves $\ll e^{O(k)}N^{O(2^{-k})}$ points of $S$; taking $k\asymp\log\log N$ gives the theorem.

Dependencies

None in the corpus. External inputs named by the paper: the determinant method bounds for rational points of bounded height on irreducible surfaces and curves (Theorems 2.2 and 2.3, from work of Castryck, Cluckers, Dittmann and Nguyen, the paper's reference [7]).

Bears on

• Problem 213: through Corollary 1.3, the theorem bounds by $O\bigl((\log N)^{O(1)}\bigr)$ the size of a set of the problem's kind lying in $[-N,N]^2$. The bound grows with $N$, so it answers neither the question nor any instance of it.
• Problem 130: through the same corollary, a clique of the problem's graph whose points lie in $[-N,N]^2$ has $O\bigl((\log N)^{O(1)}\bigr)$ vertices. This bounds no clique number, and nothing is said on the chromatic number.