On sequences of projections of the cubic lattice

TL;DR

This paper investigates sequences of lattice projections from the cubic lattice that converge to a target lattice, providing conditions for convergence rates and explicit constructions for key lattice families.

Contribution

It introduces a sufficient condition for constructing projection sequences with specific convergence rates and offers explicit methods for important lattice families.

Findings

Sequences can converge to target lattices at rate O(1/|v|^{2/n})

Explicit constructions are provided for key lattice families

A sufficient condition for convergence is established

Abstract

In this paper we study sequences of lattices which are, up to similarity, projections of $\mathbb{Z}^{n+1}$ onto a hyperplane $\bm{v}^{\perp}$, with $\bm{v} \in \mathbb{Z}^{n+1}$ and converge to a target lattice $\Lambda$ which is equivalent to an integer lattice. We show a sufficient condition to construct sequences converging at rate $O(1/ |\bm{v}|^{2/n})$ and exhibit explicit constructions for some important families of lattices.