# On the number of harmonic frames

## Simon Marshall and Shayne Waldron

## Abstract:

There is a finite number $h_{n,d}$
of tight frames of $n$ distinct vectors
for $\C^d$ which are the orbit of a
vector under a unitary action of
the cyclic group $\Z_n$.
These *cyclic harmonic frames*
(or *geometrically uniform tight frames*
are used in signal analysis
and quantum information theory,
and provide many tight frames of particular interest.
Here we investigate the
conjecture that
$h_{n,d}$
grows like $n^{d-1}$.
By using a result of Laurent which describes
the set of solutions of algebraic equations in roots of unity,
we prove the asymptotic estimate
$$ h_{n,d} \approx {n^d \over \varphi(n)}\ge n^{d-1}, \qquad n\to\infty. $$
By using a group theoretic approach,
we also give some exact formulas for $h_{n,d}$, and estimate
the number of cyclic harmonic frames up to projective unitary equivalence.

**Keywords:**
Finite tight frames,
harmonic frames,
finite abelian groups,
character theory,
Pontryagin duality,
Mordell-Lang conjecture,
projective unitary equivalence

**Math Review Classification:**
Primary 42C15, 94A12;
Secondary 11Z05, 20C15, 94A15,
20F28

**Length:** 22 pages

**Last Updated:** 22 November 2016

## Availability: