A Note on Projecting the Cubic Lattice

TL;DR

This paper demonstrates that any (n-1)-dimensional lattice can be closely approximated by projecting the integer lattice Z^n onto a carefully chosen subspace, with implications for geometric lattice problems.

Contribution

It introduces a method to approximate any (n-1)-dimensional lattice via projection from Z^n, addressing a geometric problem related to lattice cylinders.

Findings

Any (n-1)-dimensional lattice can be approximated by projection from Z^n.

The result has applications in understanding lattice cylinders with specific properties.

Provides a new perspective on lattice projection and approximation techniques.

Abstract

It is shown that, given any (n-1)-dimensional lattice L, there is a vector v in Z^n such that the projection of Z^n onto v^perp is arbitrarily close to L. The problem arises in attempting to find the largest cylinder anchored at two points of Z^n and containing no other points of Z^n.