On lattice extensions

TL;DR

This paper systematically studies lattice extensions, focusing on their geometric properties, existence, and how they affect successive minima and covering radius, with insights into deep holes of planar lattices.

Contribution

It introduces the concept of lattice extensions, proves the existence of small-determinant extensions, and analyzes their impact on lattice geometry and arithmetic properties.

Findings

Existence of small-determinant lattice extensions

Extensions preserving successive minima and covering radius

Arithmetic properties of deep holes in planar lattices

Abstract

A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.