Constructions of biangular tight frames and their relationships with equiangular tight frames

TL;DR

This paper explores biangular tight frames (BTFs), their relationship with equiangular tight frames (ETFs), and introduces new parametrizations, frameworks, and constructions linking BTFs to open problems in number theory.

Contribution

It introduces a smooth parametrization of BTFs, develops a framework for harmonic and Steiner BTFs, and constructs a biangular tight set of subspaces related to ETFs.

Findings

BTFs can be smoothly parametrized with angles transforming continuously.

A framework for harmonic and Steiner BTFs connects to open problems in prime number theory.

Constructed a biangular tight set of subspaces that embeds into an ETF.

Abstract

We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one frame angle. We demonstrate a smooth parametrization BTFs, where the corresponding frame angles transform smoothly with the parameter, which "passes through" an ETF answers two questions regarding the rigidity of BTFs. We also develop a general framework of so-called harmonic BTFs and Steiner BTFs - which includes the equiangular cases, surprisingly, the development of this framework leads to a connection with the famous open problem(s) regarding the existence of Mersenne and Fermat primes. Finally, we construct a (chordally) biangular tight set of subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.