Chojecki 2026 order growth planar sets avoiding integer

../

lemma_2: For R at least 1, M(R) is at most an absolute constant times the supremum over 0 < delta < 1/10 of delta^2 N(R, delta), and at least an absolute constant times the supremum over the same range of delta^2 N(R - 1, delta).

proposition_4: There are absolute constants A, c, s_0 > 0 such that the kernel K_s(t), the sum over k of (k + 2sk^2) e^{-sk} J_0(2 pi k t), satisfies K_s(t) <= -c (1+t)^{-1/2} whenever 0 < s < s_0 and t lies at distance at least A s from the integers.

theorem_1: For R at least 1, a measurable subset of the disk of radius R in the plane with no two distinct points at a positive integer distance has measure at most a constant times the square root of R, so the supremum M(R) of such measures is R^{1/2+o(1)} as R tends to infinity.

theorem_3: For X at least 1 and 0 < delta < 1/10, a set of points in the disk of radius X whose pairwise distances all lie at distance at least delta from the integers has at most an absolute constant times delta^{-2} X^{1/2} points.

---

Przemek Chojecki, The Order of Growth of Planar Sets Avoiding Integer Distances. preprint (ulam.ai) (2026). No notice is printed; the hosting organization's research page shows only the site footer "© 2017-2026 ULAM" and names no license (https://www.ulam.ai/research, read 2026-10-02), every other right reserved.

The three-page print names no author.

Let M(R) be the supremum of measures of measurable A inside the disk of radius R about the origin with no two distinct points at a positive integer distance. Theorem 1 (p. 1) proves M(R) << R^{1/2} for R >= 1; with a lower bound M(R) >>eps R^{1/2-eps} that the paper derives from Sarkozy's point sets (p. 3), this gives M(R) = R^{1/2+o(1)}, which the paper states settles the order-of-growth form of Erdos Problem #953. Lemma 2 (p. 1) shows that, for R >= 1, M(R) is bounded above and below, up to absolute constants, by the suprema over 0 < delta < 1/10 of delta^2 N(R,delta) and delta^2 N(R-1,delta) respectively, where N(X,delta) is the largest size of a point set in the disk of radius X whose pairwise distances stay at least delta from the integers. Theorem 3 (p. 1) supplies N(X,delta) << delta^{-2} X^{1/2} for X >= 1 and 0 < delta < 1/10; the paper attributes a bound of this type for each fixed delta, without uniform dependence, to Konyagin. The tool is the positive-definite Poisson-Bessel kernel K_s(t) = sum{k>=1} (k + 2 s k^2) e^{-sk} J_0(2 pi k t) of Section 2, with K_s(0) << s^{-2}; Proposition 4 (p. 2) gives absolute A, c, s_0 > 0 with K_s(t) <= -c (1+t)^{-1/2} whenever 0 < s < s_0 and the distance from t to the integers is at least A s, and its proof shows every term of the Poisson expansion is non-positive there. Positive definiteness then gives Theorem 3 (p. 3).

Read status: claims checked for Theorem 1, Lemma 2, Theorem 3 and Proposition 4 with the definitions they use, each read clause by clause on the printed pages. The proofs (pp. 1--3) were read for structure only; none was checked, and nothing here is independently reviewed.

Contents

• § 1, Statement and reduction (p. 1): the definitions of M(R) and N(X,delta), Theorem 1, Lemma 2 with its proof, and Theorem 3.
• § 2, The kernel (pp. 2--3): the kernel (2.1), its positive definiteness and the diagonal bound (2.2), Proposition 4 and its proof.
• § 3, Completion of the proof (p. 3): the proofs of Theorem 3 and Theorem 1, and the references.

Source: https://www.ulam.ai/research/erdos953-short.pdf.

Bears on. #953: Theorem 1 (p. 1) proves the upper bound M(R) << R^{1/2} for R >= 1 on the largest measure the problem asks about, and with the lower bound the paper derives from Sarkozy's construction this gives M(R) = R^{1/2+o(1)}; it does not decide whether M(R) has order exactly R^{1/2}. Lemma 2 and Theorem 3 (p. 1) and Proposition 4 (p. 2) are the steps of its proof. #466: Theorem 3 (p. 1) bounds from above the problem's quantity N(X,delta), uniformly for 0 < delta < 1/10; it does not decide whether N(X,delta) tends to infinity, which is the problem's question.

Results.

• Theorem 1 (p. 1): M(R) << R^{1/2} for R >= 1, hence M(R) = R^{1/2+o(1)} as R tends to infinity.
• Lemma 2 (p. 1): for R >= 1, M(R) << sup over 0 < delta < 1/10 of delta^2 N(R,delta), and M(R) >> sup over 0 < delta < 1/10 of delta^2 N(R-1,delta).
• Theorem 3 (p. 1): N(X,delta) << delta^{-2} X^{1/2} for X >= 1 and 0 < delta < 1/10.
• Proposition 4 (p. 2): for absolute A, c, s_0 > 0, K_s(t) <= -c (1+t)^{-1/2} whenever 0 < s < s_0 and the distance from t to the integers is at least A s.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.