Lattices from tight equiangular frames

TL;DR

This paper investigates when integer linear combinations of vectors from tight equiangular frames form lattices, showing conditions under which they do or do not, and analyzing specific low-dimensional cases.

Contribution

It establishes new criteria for when these sets form lattices, including irrational cosine cases, and provides detailed analysis of low-dimensional examples.

Findings

Set is not a lattice if the cosine of the frame angle is irrational.

Set forms a lattice when n = k+1.

A (7,28) frame generates a strongly eutactic and perfect lattice.

Abstract

We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular $(k,n)$ frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the $k$-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for $n = k+1$ and that there are infinitely many $k$ such that a lattice emerges for $n = 2k$. We dispose of all cases in dimensions $k$ at most $9$. In particular, we show that a $(7,28)$ frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect.