Ambrus 2020 density estimates 1 avoiding sets via

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theorem_1: Ambrus and Matolcsi's theorem that every Lebesgue measurable planar set with no two points at distance 1 has upper density at most 0.25442, so m_1(R^2) <= 0.25442.

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Gergely Ambrus, Máté Matolcsi, Density estimates of 1-avoiding sets via higher order correlations. Discrete Comput. Geom. 67 (2022), 1245-1256. DOI 10.1007/s00454-020-00263-3. arXiv:1809.05453. The copy read for this card is arXiv v3 (20 Oct 2020); page numbers below are that copy's.

Theorem 1 (p. 2) bounds the upper density of every Lebesgue measurable planar set with no two points at distance 1 by 0.25442, improving the previous best 0.25646 of Bellitto, Pecher and Sedillot. The method is Fourier analysis plus linear programming, following Keleti-Matolcsi-Oliveira Filho-Ruzsa, with the new ingredient of linear constraints on the autocorrelation function derived from triple-order (three-point) correlations of the set (Lemmas 3 and 4 and the constraint (CT), pp. 4-6), a concept the abstract calls not previously studied. The bound stays above 1/4, so it does not settle Erdos's conjecture that m_1(R^2) < 1/4; the paper records Croft's 0.22936 as the best lower bound. The later bound 0.2470 of Ambrus, Csiszarik, Matolcsi, Varga and Zsamboki (card) improves on it. The result concerns measurable 1-avoiding sets only and gives no bound on the chromatic number chi(R^2).

Source: https://arxiv.org/abs/1809.05453. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1809.05453), every other right reserved.

Read status. Claims checked: Theorem 1 and the definitions it uses were read clause by clause on the printed pages, and the outline of its proof (Lemmas 1-4, the constraint (CT) and Proposition 1, pp. 3-8) was read. The numerical certificate (Section 5, pp. 8-9, and Tables 1-3, p. 10) was not recomputed.

Bears on. #508: the paper names the chromatic number of the plane as a related question (p. 2) and proves nothing about it; Theorem 1 bounds the density of a measurable set with no unit distance, which leaves the bounds on chi(R^2) where they stood.

Results. Theorem 1 (p. 2). Lemmas 3 and 4 (pp. 4-5), the constraint (CT) (p. 6) and Proposition 1 (p. 8) are steps of its proof, summarized on its page.

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