Cohen 2024 clustering typical unit distance avoiding sets

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theorem_1_3: States that for some gamma > 0, every periodic planar set A with s(1;A) at most gamma and density at least m_1(R^2) - gamma has normalized pair correlation s(1.96;A) at least 1 + gamma.

theorem_1_4: States that for all K > K_0 and N > N_0(K), a uniformly random 1-avoiding set among those locally constant at scale 1/N and periodic modulo K Z^2 has s(1.96;A) at least 1 + gamma/2 with high probability.

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Alex Cohen, Nitya Mani, Clustering in typical unit-distance avoiding sets. arXiv preprint (2024). arXiv:2407.05071. The version read is v1, dated July 9, 2024 on its first page (20 pages); labels and pages below are those of v1.

The paper rigorously establishes the clustering phenomenon observed in dense 1-avoiding planar sets, meaning sets containing no two points exactly distance 1 apart. Theorem 1.3 shows there is gamma > 0 such that any periodic set A in the plane with normalized pair correlation s(1;A) <= gamma and density at least m_1(R^2) - gamma satisfies s(1.96;A) >= 1 + gamma, so nearly extremal 1-avoiding sets have over-represented pairs at distance about 2 (the paper uses 1.96 rather than 2 for technical reasons, says its proof gives an explicit numerical value of gamma without printing it, and proves the theorem under the potentially weaker hypothesis that the density is at least that of Croft's set minus gamma). Theorem 1.4 extends this from extremal to typical sets: for all K > K_0 and N > N_0(K), a uniformly random 1-avoiding set among the 1/N-constant K-periodic ones satisfies s(1.96;A) >= 1 + gamma/2 with high probability. It is proved by the graph container method, whose main input is a supersaturation principle for unit distances in the plane, due to Bourgain and developed by Bukh, resting on compactness of the functional A maps to f°(1;A) and the curvature of the Euclidean norm. The technique builds on the linear programming approach to the density of 1-avoiding sets, working with the radialized autocorrelation, i.e. pair correlation, function. For problem 1070 the paper bears only through the density m_1(R^2), which bounds f(n) from below by f(n) >= m_1 n: it repeats Croft's tortoise construction of density about 0.22936 and the Ambrus-Csiszárik-Matolcsi-Varga-Zsámboki upper bound of 0.2470 on m_1 (Theorem 1.1), without changing either, and its clustering results give no new bound on f(n).

Source: https://arxiv.org/abs/2407.05071. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2407.05071), every other right reserved.

Results.

• Theorem 1.3 (p. 3): there is $\gamma>0$ such that every periodic $A\subset\mathbb R^2$ with $s(1;A)\le\gamma$ and $\delta(A)\ge m_1(\mathbb R^2)-\gamma$ has $s(1.96;A)\ge1+\gamma$.
• Theorem 1.4 (p. 4), with its graph form Theorem 3.3 (p. 9): for all $K>K_0$ and $N>N_0(K)$, a uniformly random 1-avoiding set among those locally constant at scale $1/N$ and $K$-periodic has $s(1.96;A)\ge1+\gamma/2$ with high probability.

Cited, not proved here: Theorem 1.1 (p. 1), that every Lebesgue measurable 1-avoiding planar set has upper density at most $0.2470$, is the result of Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki; Croft's tortoises on a hexagonal lattice (pp. 1--2) give the densest known 1-avoiding planar set, of density about $0.22936$ at $x$ about $0.96553$, above the disc-packing value $\pi/(8\sqrt3)\approx0.2267$. The paper also poses Question 5.1 (p. 17), whether some set with block structure has density $m_1$.

Read status. Claims checked for Theorems 1.3 and 1.4, read clause by clause on the print; the proofs were read for their structure, and the numerical verification of the witness function (Section 4.1) was not rerun.

Bears on

• Problem 1070: background only. The problem's $f(n)$ is at least $m_1(\mathbb R^2)\,n$. The paper restates the known bounds on $m_1(\mathbb R^2)$, at least the density of Croft's construction (about $0.22936$) and at most $0.2470$ (Theorem 1.1, cited), without changing them. Theorems 1.3 and 1.4 describe the pair correlation of dense sets with few unit-distance pairs and of typical discretized 1-avoiding sets; none of the paper's results gives a bound on $f(n)$.

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