On existence of integral point sets and their diameter bounds

Nikolai Avdeev

paper

On existence of integral point sets and their diameter bounds

arXiv:1906.11926

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 $d$ .