On sequences of projections of the cubic lattice

Abstract

In this paper we study sequences of lattices which are, up to similarity, projections of ${\mathbb{Z}}^{n+1}$ onto a hyperplane ${\mathbf{v}}^{\perp }$, with $\mathbf{v}\in {\mathbb{Z}}^{n+1}$ 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 $O(1/|\mathbf{v}{|}^{2/n})$ and exhibit explicit constructions for some important families of lattices.

arXiv comment16 pages, 5 figures