Lattices from tight equiangular frames
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2016
- DOI:
- 10.48550/arXiv.1607.05350
- arXiv:
- arXiv:1607.05350
- Bibcode:
- 2016arXiv160705350B
- Keywords:
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- Mathematics - Functional Analysis;
- 15B35;
- 05B30;
- 11H06;
- 42C15;
- 52C07
- E-Print:
- 25 pages