Year: 2026
Type: preprint
Abstract: We provide a sharp estimate for the asymptotic number of lattice zonotopes inscribed in [0, n] d as n tends to infinity. Our estimate refines the logarithmic equivalent established by B{á}r{á}ny, 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 a polynomial of degree d in n 1/(d+1) , plus a term involving the non-trivial zeros of the Riemann zeta function. Our analysis is based on a correspondence between power sums over primitive vectors (vectors with coprime coordinates) and Eulerian polynomials. We also analyse some combinatorial properties of lattice zonotopes. In particular, we compute the asymptotic mean and variance of the diameter of the graph of a random lattice zonotope inscribed in [0, n] d as n goes to infinity, and show that this diameter satisfies a central limit theorem.
Source: arXiv (Cornell University)
Authors: Bodini, Olivier, Buffière, Théophile
Institution: Laboratoire Analyse, Géométrie et Applications
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224966956