Ruhland 2025 no new lower bound density planar

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correction_p1: The author reports a severe error in the two-parameter minimization of the earlier versions and states that, after correction, none of the sets of constant diameter investigated in the paper gives a new lower bound for the density of planar sets avoiding unit distances.

definition_1: A planar set has constant diameter when every boundary point has the same diameter, the supremum of its distances to points of the set; the paper uses such sets as the uncut shapes of its tortoises.

equation_4_5: The paper expands the area of the cut sets of constant diameter in its family to second order in the parameter, printing a positive quadratic coefficient that conflicts with the outcome its Introduction reports.

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Helmut Ruhland, No new lower bound for the density of planar sets avoiding unit distances. arXiv preprint (2025). arXiv:2408.10076. The copy read for this card is arXiv:2408.10076v4 (18 June 2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2408.10076), every other right reserved.

The paper studies the maximal density m_1 of a planar set avoiding unit distances, where Croft's 1967 tortoise construction gives the lower bound 0.22936 and Ambrus et al. give the upper bound 0.2470 for periodic sets. Earlier versions of this article claimed a construction of planar sets of constant diameter with density higher than Croft's; in this version the author reports a severe error in the former two-parameter minimization (former (F.7) of Appendix F), found by comparing theoretical values with a numerical implementation. After the correction, no set of constant diameter that the paper examines gives a new lower bound, and the article was retitled rather than withdrawn. Sections 2-4 define a one-parameter family D_eps of sets of constant diameter 2, build 2-avoiding sets S_eps from them, and expand the tortoise areas as power series. The outline in the introduction (p. 1) says that this expansion shows the density of S_eps to be below that of Croft's S_0 for small eps not equal to 0, but the end of Section 4 (equation (4.5), p. 6) and the Conclusion (p. 6) still state the earlier versions' opposite reading: a positive eps^2 coefficient in the tortoise area, so that Croft's density is a local minimum and S_eps has density at least Croft's for small eps. For problem 1070 the paper is therefore a negative record: it is the author's own retraction and supplies no new bound on the density of unit-distance-avoiding planar sets.

Source: https://arxiv.org/abs/2408.10076.

Bears on.

• #1070: the problem page records f(n) >= m_1 n (Larman and Rogers) with Croft's m_1 >= 0.22936. The paper withdraws its author's earlier claim to improve that lower bound on m_1 and proves no bound on m_1 or on f(n).

Results.

• Correction (p. 1): the former two-parameter minimization (former F.7) contained a severe error, and after its correction none of the sets of constant diameter investigated gives a new lower bound.
• Definition 1 (p. 2): sets of constant diameter, every boundary point having the same diameter (supremum of distances to points of the set); the paper remarks that such sets are locally area-maximal and uses them as tortoises before cutting.
• Equation (4.5) (p. 6): the tortoise area expanded to second order in eps, printed with a positive eps^2 coefficient 0.0013926262 and read on p. 6 as a local minimum of the density at Croft's S_0; the Introduction (p. 1) states the opposite outcome after the correction.

Read status. Claims checked: the statements on the result pages were read clause by clause on the printed pages of v4; no computation or proof was checked.

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