Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 953

claims/: The 1 claim page of Problem 953, one per claimant's result; the problem's standing derives from them.

Statement. Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert \not\in \mathbb{Z}$ for any distinct $a,b\in A$ . How large can the measure of $A$ be?

Formulation. The question "how large" asks for the size of $M(r)$ , the supremum of the measure of such a set $A$ , as $r$ tends to infinity. The bounds $M(r)\gg_\varepsilon r^{1/2-\varepsilon}$ and $M(r)\ll r^{1/2}$ settle the exponent, $M(r)=r^{1/2+o(1)}$ , but leave open the constant-factor question: whether $M(r)$ has order exactly $r^{1/2}$ , that is, whether the factor $r^{o(1)}$ between the two bounds can be removed. The problem stays open until that question is settled.

Status. Open. The site keeps the label OPEN, and its curator has not commented on Chojecki's upper bound. That bound, which with the lower bound the site credits settles the exponent $1/2$ , is an accepted partial result recorded on its claim page; the constant-factor question is open.

Source. erdosproblems.com/953, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #953, https://www.erdosproblems.com/953.

References.

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

Formalization. The formal-conjectures catalog has no statement file for the problem. Allen Hart's Lean proof of the upper bound is linked from the claim page and has not been built here.

Current assessment

Open; the exponent $1/2$ is settled. Trivially $M(r)=O(r)$ . Koizumi and Kovač observed on the site's discussion thread (comments of 9 August 2025 and 1 February 2026, which the site credits) that Sárközy's lower bound on Problem 466, recorded on its claim page, adapts to $M(r)\gg_\varepsilon r^{1/2-\varepsilon}$ for every $\varepsilon>0$ , by placing small discs at a robust point set. Chojecki's notes of April 2026 prove $M(r)\ll r^{1/2}$ for $r\geq1$ with a positive-definite Poisson-Bessel kernel, so $M(r)=r^{1/2+o(1)}$ ; Vjekoslav Kovač wrote on the thread on 8 May 2026 that the proof is correct, and the result is recorded as an accepted partial result on its claim page. The notes and Sothanaphan's streamlined write-up are carded at chojecki_2026_poisson_bessel_kernel_bound_planar_sets, chojecki_2026_order_growth_planar_sets_avoiding_integer and sothanaphan_2026_compact_poissonbessel_proof_integer_distance_free. The remaining gap, and the open question, is the factor $r^{o(1)}$ in the lower bound: whether $M(r)$ has order exactly $r^{1/2}$ . Search scope: the site's problem page and discussion thread, the claimant's notes, the Lean repository linked from the thread and the formal-conjectures catalog; no forum proof claim, release item or lead names the problem.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (22)

Chojecki's Poisson-Bessel bound for integer-distance-free sets problems/distance_problems/E0953/claims Distance Problems discrete_geometry/erdos_1982_my_favourite_problems_which_recently_have Distance Problems distance_problems/chojecki_2026_order_growth_planar_sets_avoiding_integer Lemma 2 (p. 1): the measure M(R) is comparable to the robust point counts delta^2 N(R, delta)Proposition 4 (p. 2): the Poisson-Bessel kernel K_s(t) is at most -c(1+t)^{-1/2} away from the integers Theorem 1 (p. 1): planar sets avoiding integer distances in a disk of radius R have measure O(R^{1/2})Theorem 3 (p. 1): N(X, delta) << delta^{-2} X^{1/2} uniformly in delta distance_problems/chojecki_2026_poisson_bessel_kernel_bound_planar_sets Lemma 3.1 (p. 2): the Poisson-Bessel kernel is positive definite on the plane, with K_s(0) << s^{-2}Proposition 2.1 (p. 2): c sup delta^2 N(R-1, delta) <= M(R) <= C sup delta^2 N(R, delta)Proposition 4.2 (p. 5): K_s(t) <= -c(1+t)^{-1/2} when 0 < s < s_0 and dist(t, Z) >= As Theorem 1.1 (p. 1): M(R) << R^{1/2} for all R >= 1, hence M(R) = R^{1/2+o(1)}Theorem 1.2 (p. 1): N(X, delta) <= C delta^{-2} X^{1/2} for X >= 1 and 0 < delta < 1/10 distance_problems/sothanaphan_2026_compact_poissonbessel_proof_integer_distance_free Lemma 1.1 (p. 2): the Poisson-Bessel kernel is positive definite, bounded by K_s(0) << s^{-2}, and expands into integer-centred terms Lemma 1.2 (p. 2): sign and size of one Poisson term T_m(s,t)Proposition 1.3 (p. 3): K_s(t) <= -c(1+t)^{-1/2} away from the integers Theorem 2.1 (p. 4): M(R) << R^{1/2} for R >= 1 integer_sequences/erdos_1972_extremal_problems_number_theory

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