Primitive point packing

TL;DR

This paper investigates the maximum number of primitive lattice points with bounded coordinate sums, providing solutions to a packing problem that connects geometry, number theory, and combinatorics, with implications for lattice zonotopes.

Contribution

It offers a solution to the primitive point packing problem and derives explicit formulas for the maximum diameter of lattice zonotopes within hypercubes.

Findings

Explicit expression for the maximum diameter of lattice zonotopes.

Solution to the packing problem for primitive points with bounded coordinate sums.

Conjecture on the maximum diameter of lattice polytopes in hypercubes.

Abstract

A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the primitive points within the lattice, the ones whose first non-zero coordinate is positive. We solve the packing problem that asks for the largest possible number of such points whose absolute values of any given coordinate sum to at most a fixed integer $k$. We present several consequences of this result at the intersection of geometry, number theory, and combinatorics. In particular, we obtain an explicit expression for the largest possible diameter of a lattice zonotope contained in the hypercube $[0,k]^d$ and, conjecturally of any lattice polytope in that hypercube.