Ascher 2019 erdos ulam problem lang s conjecture

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corollary_1_2: The contrapositive of Theorem 1.1: if there are rational distance sets in general position of cardinality larger than any fixed constant, then Lang's Conjecture is false.

definition_p1: Ascher, Braune and Turchet call an n-point subset of the plane in general position when no n-4 of its points lie on a line and no n-3 on a circle; read literally it fails for 4 <= n <= 6, and at n = 7 it is no three collinear and no four concyclic.

proposition_3_6: Assuming Lang's Conjecture, every rational distance set lies in a proper Zariski-closed subset of the complex projective plane, which can be chosen with the degrees of its irreducible components summing to at most one integer d valid for all rational distance sets.

proposition_4_4: A normal projective complex variety of dimension d with Q-Cartier ample canonical divisor, whose non-canonical singularities are ordinary multiple points, is of general type when K_X^d exceeds the sum over those points of the d-th power of the absolute discrepancy times the multiplicity.

proposition_5_1: For m distinct points of the plane, the surface in P^(2+m) cut out by the quadrics r_j^2 = (x - a_j z)^2 + (y - b_j z)^2 is of general type when m is at least 4; for m = 4 this is Tao's result.

theorem_1_1: Ascher, Braune and Turchet's main theorem: assuming Lang's Conjecture, one constant bounds the cardinality of every rational distance set in the plane that is in general position in the paper's sense.

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Kenneth Ascher, Lucas Braune, Amos Turchet, The Erdős-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2019); published as The Erdős-Ulam problem, Lang's conjecture and uniformity, Bull. Lond. Math. Soc. 52 (2020), no. 6, 1053--1063, doi:10.1112/blms.12381. The copy read for this card is arXiv:1901.02616v2 (17 August 2020); the labels and pages cited below are those of that version. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1901.02616), every other right reserved.

A rational distance set is a set of points of R^2 whose pairwise distances are all rational; the paper's Theorem 1.1 (p. 1) shows that, assuming Lang's Conjecture (Conjecture 2.2), one constant bounds the size of every rational distance set in general position, where an n-point set is in general position when no n-4 of its points lie on a line and no n-3 on a circle. Corollary 1.2 restates this contrapositively: if arbitrarily large rational distance sets in general position exist, Lang's Conjecture fails. The method lifts points of such sets to rational points on curves and surfaces of general type (following Solymosi-de Zeeuw and Tao) and then applies the uniformity theorems of Caporaso-Harris-Mazur for curves and Hassett for surfaces. Along the way Proposition 4.4 (p. 7) gives a criterion for a normal projective variety with ample canonical divisor, whose non-canonical singularities are ordinary multiple points, to be of general type; through Proposition 5.1 it yields the general-type statement for the singular surfaces cut out by four or more quadrics that the proof needs, which for four quadrics in P^6 is Tao's result. For the Erdős-Ulam problem (Question 1, p. 1: is some rational distance set dense in the Euclidean topology of R^2) the paper recalls that Shaffaf and Tao independently showed that Lang's Conjecture implies the answer no, indeed that no rational distance set is Zariski dense; its Proposition 3.6 (p. 5) adds a bound, uniform over all rational distance sets, on the total degree of a proper algebraic set containing one, and its abstract presents the main theorem as generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf and Tao. For Erdős' question on seven points in general position with rational distances it notes the affirmative answer of Kreisel and Kurz, says the authors know of no rational distance set in general position with more than seven points, and notes that for more than seven points its notion of general position is strictly weaker than the no-three-collinear, no-four-concyclic notion of Kreisel and Kurz (p. 2).

Source: https://arxiv.org/abs/1901.02616.

Bears on. #212: Proposition 3.6 (p. 5) puts every rational distance set, assuming Lang's Conjecture, inside a proper algebraic subset of the plane, so under that unproven conjecture none is dense and the question would be answered no; this conditional answer is the earlier result of Shaffaf and Tao, which the paper cites (p. 1), and the proposition adds the uniform degree bound. #213: a set of n >= 7 points with no three on a line, no four on a circle and integer distances is a rational distance set in general position in the paper's sense, so Theorem 1.1 (p. 1) bounds the size of such sets assuming Lang's Conjecture, and under that unproven conjecture the question, read as asking for every n >= 4, would be answered no; no value of the bound is given, and the theorem decides no instance.

Results. Labels and pages are those of arXiv:1901.02616v2.

• Definition (p. 1, unnumbered): general position, no n-4 points on a line and no n-3 on a circle.
• Theorem 1.1 (p. 1): under Lang's Conjecture, rational distance sets in general position have bounded size.
• Corollary 1.2 (p. 1): arbitrarily large such sets would refute Lang's Conjecture.
• Proposition 3.6 (p. 5): under Lang's Conjecture, rational distance sets lie in proper algebraic subsets of the plane of uniformly bounded total degree.
• Proposition 4.4 (p. 7): a general-type criterion for varieties whose non-canonical singularities are ordinary multiple points.
• Proposition 5.1 (p. 8): the surface cut out by m >= 4 distance quadrics is of general type.

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