A solution of the Erdos-Ulam problem on rational distance sets assuming the Bombieri-Lang conjecture

TL;DR

This paper links the Erdős-Ulam conjecture on dense rational distance sets in the plane to the Bombieri-Lang conjecture, showing that assuming the latter implies the former, and characterizes large rational distance sets as mostly on a line or circle.

Contribution

It introduces a new algebraic surface called a distance surface and proves it is of general type under certain conditions, connecting the problem to deep conjectures in arithmetic geometry.

Findings

Distance surfaces associated with rational distance sets are of general type.

Assuming the Bombieri-Lang conjecture, dense rational distance sets cannot exist.

Large rational distance sets are mostly contained in a line or circle, with at most four or three exceptions.

Abstract

A rational distance set in the plane is a point set which has the property that all pairwise distances between its points are rational. Erd\H os and Ulam conjectured in 1945 that there is no dense rational distance set in the plane. In this paper we associate an algebraic surface in $\mathbb{P}^3$, that we call a distance surface, to any finite rational distance set in the plane. Under a mild condition, we prove that a distance surface is always a surface of general type. From this, we deduce that the Bombieri-Lang conjecture in arithmetic algebraic geometry (restricted to the classes of surfaces) implies an answer to the Erd\H os-Ulam problem. Combined with the results of Solymosi and de Zeeuw, our proofs lead to the following stronger statement: for $S$ a rational distance set with infinitely many points, we have Either, all but at most four points of $S$ are on a line, Or, all but at most three points of $S$ are on a circle.