On existence of integral point sets and their diameter bounds
Nikolai Avdeev
TL;DR
The paper studies integral point sets in Euclidean spaces, focusing on lower bounds for the diameter of planar sets and constructive existence results in higher dimensions. By linking planar diameter problems to the Point Packing in a Square problem, it achieves a sharper linear lower bound with a constant around 0.4653 (thus exceeding 5/11 for large n) and establishes that the dual quantity l(m,n) equals 1 for all admissible m,n. It then provides a general construction ensuring the existence of n-point integral point sets in m dimensions with a prescribed distance d between two points, and discusses prime and facher sets, blowing-up techniques, and implications for higher dimensions. The work also outlines several open questions and conjectures regarding optimal bounds, geometric containment, and unit-distance occurrences.
Abstract
A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower bound for diameter of planar integral point sets. This improvement takes into account some results related to the Point Packing in a Square problem. Then for arbitrary integers $m \geq 2$, $n \geq m+1$, $d \geq 1$ we give a construction of an integral point set $M$ of $n$ points in $m$-dimensional Euclidean space, where $M$ contains points $M_1$ and $M_2$ such that distance between $M_1$ and $M_2$ is exactly $d$.