On integer distance sets

TL;DR

This paper introduces a new approach to analyze integer distance sets in the Euclidean plane, revealing their structural dichotomy and providing bounds on their size and configuration.

Contribution

It establishes that integer distance sets are either very sparse or mostly collinear or circular, and derives bounds on their diameter and size under certain conditions.

Findings

Integer distance sets are either sparse or mostly on a line or circle.

Derived near-optimal lower bounds on the diameter of non-collinear sets.

Provided strong upper bounds on the size of sets with no three collinear points and no four cocircular points.

Abstract

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an exceedingly small proportion of its points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any non-collinear integer distance set of size $n$ and a strong upper bound on the size of any integer distance set in $[-N,N]^2$ with no three points on a line and no four points on a circle.