Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules

TL;DR

This paper reviews methods from algebra and number theory to count and classify sublattices and $bZ$-modules, focusing on their enumeration, asymptotic behavior, and applications in crystallography.

Contribution

It introduces algebraic and analytic techniques for enumerating and analyzing sublattices and $bZ$-modules, extending results to higher-dimensional modules.

Findings

Enumeration formulas for sublattices based on index

Asymptotic growth rates of counting functions

Applications to crystallography and lattice theory

Abstract

In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to $\mathbb{Z}$-modules embedded in $\mathbb{R}^d$.