Point Counts of D_k and Some A_k and E_k Integer Lattices Inside Hypercubes

TL;DR

This paper counts lattice points of D_k, A_k, and E_k integer lattices within hypercubes, providing polynomial formulas based on the hypercube edge length, to analyze their geometric properties.

Contribution

It introduces a method to count lattice points inside hypercubes for D_k, A_k, and E_k lattices, with polynomial formulas depending on the hypercube size.

Findings

Derived polynomial formulas for lattice point counts

Analyzed lattice point distributions within hypercubes

Extended counting methods to different lattice types

Abstract

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm in the host lattice. The results assume polynomial format as a function of the hypercube edge length.