[Paper Review] On the group of rational spectral units with finite order

Emmanuel Amiot|ArXiv.org|Jul 5, 2009

Mathematical Analysis and Transform Methods1 references3 citations

TL;DR

This paper characterizes the subgroup of rational spectral units with finite order in the context of homometric sets on cyclic groups. Using Fourier analysis and eigenvalue relations under the action of the multiplicative group modulo n, it proves that such spectral units are completely determined by their eigenvalues being roots of unity, with constraints depending on the divisors of n and the parity of n.

ABSTRACT

The problem of phase retrieval is a difficult one which remains far from solved. Two homometric sets are always connected by way of a convolution product by some spectral unit, though not necessarily in a unique way. Here we elucidate one small aspect, the subgroup of spectral units with finite order. Its elements are completely characterized by relations between their eigenvalues. This sheds some light on the beltway problem.

Motivation & Objective

• To characterize the subgroup of rational spectral units with finite order in the context of homometric sets on cyclic groups.
• To understand the algebraic structure of spectral units that preserve interval content under convolution.
• To determine the conditions under which such spectral units have rational coefficients and finite order.
• To clarify the role of eigenvalues and their relations under the action of Z_n^* in defining these units.
• To connect the beltway problem and Z-relation in music theory with algebraic number theory via spectral units.

Proposed method

• Uses the isomorphism between circulant matrices, convolution algebras, and diagonal matrices via the Fourier transform.
• Applies the Fourier matrix Ω to diagonalize circulant matrices, linking spectral units to diagonal matrices with unimodular eigenvalues.
• Analyzes the action of the multiplicative group Z_n^* on eigenvalues to determine orbit structures and constraints.
• Employs the Chinese Remainder Theorem and properties of differences in Z_n^* to derive conditions on eigenvalues.
• Uses the Bezout identity to show that each j ∈ Z_n is associated with a divisor d = gcd(n,j), reducing the problem to divisor representatives.
• Applies Lemma 6, showing that Δ_n = Z_n^* - Z_n^* equals Z_n when n is odd and 2Z_n when n is even, to constrain eigenvalue orders.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a spectral unit with rational coefficients to have finite order?
• RQ2 How do the eigenvalues of a rational spectral unit of finite order relate to roots of unity and the structure of Z_n?
• RQ3 What role do the divisors of n and the parity of n play in determining the possible eigenvalues of such spectral units?
• RQ4 Can the finite-order rational spectral units be fully characterized by their action on the orbits of Z_n^*?
• RQ5 How does the structure of the multiplicative group Z_n^* constrain the eigenvalue relations in finite-order rational spectral units?

Key findings

• All rational spectral units of finite order are completely determined by their eigenvalues being roots of unity, with constraints depending on the divisors of n.
• When n is even, each eigenvalue ξ_j for j dividing n must be a root of unity of order dividing n or 2n, depending on the 2-adic valuation of n/j.
• When n is odd, ξ_j is a root of unity of order dividing n if it is a n-th root, or of order dividing 2n if it is a primitive 2n-th root.
• For any divisor j of n, the eigenvalue ξ_j is constrained by the existence of k, k' ∈ Z_n^* such that k' - k = n/j (odd case) or 2n/j (even case), leading to ξ_j^{n/j} = 1 or ξ_j^{2n/j} = 1.
• If an eigenvalue ξ_j is not a n-th root of unity, then -ξ_j must be a n-th root of unity, which only occurs when n/j is odd.
• The entire spectral unit is determined by choosing, for each divisor d of n, a root of unity of order dividing n or 2n, depending on the parity of n/d.