The asymptotic number of lattice zonotopes in a hypercube

TL;DR

This paper derives a precise asymptotic estimate for the number of lattice zonotopes within a hypercube as the size parameter grows, refining previous logarithmic estimates and revealing deep connections to number theory and combinatorics.

Contribution

It provides the first sharp asymptotic formula for lattice zonotopes in a hypercube, involving Riemann zeta zeros and Eulerian polynomials, and analyzes their combinatorial properties.

Findings

Asymptotic count involves Riemann zeta function and non-trivial zeros.

Refines previous logarithmic estimates for lattice zonotopes.

Analyzes the first moment of the polyhedral graph's diameter.

Abstract

We provide a sharp estimate for the asymptotic number of lattice zonotopes, inscribed in $[0,n ]^d$ when $n$ tends to infinity. Our estimate refines the logarithmic equivalent established by Barany, Bureaux, and Lund when the sum of the generators of the zonotope is prescribed. As we shall see, the exponential part of our estimate is composed of a polynomial of degree $d$ in $n^{1/(d+1)}$, and involves Riemann's zeta function and its non-trivial zeros. %Our analysis is based on a mapping between sums of coprime numbers and Eulerian polynomials. We also analyze some combinatorial properties of lattice zonotopes. In particular, we provide the first moment of the polyhedral graph asymptotic diameter when $n$ goes to infinity.