The Erdős–Ulam problem, Lang's conjecture and uniformity

Abstract

A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

“…Conjecture 1.3 has a wide range of important consequences; we mention for example the abc conjecture of Masser-Oesterlé [33,Conjecture 3], the Bombieri-Lang conjecture [24, Conjecture F.5.2.1] and the Lang-Vojta conjecture [24,Conjecture F.5.3.6]. We refer to [1,2,4,25] for further consequences.…”

“…Remark 1. 1 In the statement of Theorem A, given a projective embedding of X, the degree of σ (C) is bounded by a constant C multiplied by max {1, χ S (C)}, where the constant C depends only on the geometric data of the finite map π and its ramification. The dependence on the curve appears only in the Euler characteristic.…”

On the Number of Perfect Triangles with a Fixed Angle

“…In particular, this implies that X 12 is a smooth curve. 2 Now by applying Lemma 2.3 (see [8,Lem. 3]) X 12 is an irreducible curve with genus 5.…”

“…In the same circle of ideas, the weak Lang conjecture was used [12] to show that if S is a rational distance set of R 2 that intersects any line in only finitely many points, then there is a uniform bound on the cardinality of the intersection of S with any line. Recently, Ascher et al [2] considered rational distance sets S ⊂ R 2 such that no line contains all but at most four points of S, and no circle contains all but at most three points of S. They showed by assuming the weak Lang conjecture that there exists a uniform bound on the cardinality of such sets S.…”

The Uniformity Conjecture in Additive Combinatorics

“…We are going to use a corollary of the conjecture, the Uniformity Conjecture [7] for special curves. There are nice applications of the Bombieri-Lang conjecture in combinatorics, for example it would imply the Erdős-Ulam conjecture, that there is no everywhere dense subset of the real plane where the distance of any two points is a rational number [11,24,26,20,13].…”