Geometry and combinatoric of Minkowski--Voronoi 3-dimesional continued fractions

TL;DR

This paper explores the combinatorial structure of 3D Minkowski-Voronoi continued fractions, proving asymptotic stability in certain lattice families and explicitly constructing complexes for White's lattices.

Contribution

It introduces new results on the stability of Minkowski-Voronoi complexes and provides explicit constructions and hypotheses for more complex cases.

Findings

Proved asymptotic stability of Minkowski-Voronoi complexes in specific lattice families.

Explicitly constructed complexes for White's rank-1 lattices.

Hypothetic description for complex lattice settings.

Abstract

In this paper we investigate the combinatorial structure of 3-dimensional Minkowski-Voronoi continued fractions. Our main goal is to prove the asymptotic stability of Minkowski-Voronoi complexes in special two-parametric families of rank-1 lattices. In addition we construct explicitly the complexes for the case of White's rank-1 lattices and provide with a hypothetic description in a more complicated settings.