The Spectral Structure of Fractional Fields
From matrices to eigenvalues
The cross-alignment matrix A(p) encodes the full pairwise coherence of the fractional field. For digit-partitioning primes, it's the identity. For non-DP primes and composites, it has rich spectral structure. This paper proves that the eigenvalues have an explicit harmonic-analytic description.
The circulant theorem
For primes where the base b is a primitive root (ord_p(b) = p − 1), every fraction k/p is a cyclic rotation of 1/p. The cross-alignment matrix is therefore a circulant matrix, and its eigenvalues are the discrete Fourier transform of the repetend's cyclic autocorrelation:
$$\lambda_j = \frac{1}{L} \sum_{\ell=0}^{L-1} R(\ell), e^{2\pi i j\ell / L}$$
The eigenvalues are character values. The alignment spectrum IS the power spectral density of the base repetend.
Special cases
For DP primes: the autocorrelation is R(0) = L, R(ℓ) = 0 for ℓ > 0. The DFT is flat: all eigenvalues equal 1. The identity matrix.
For two-coset primes (like p = 13): the matrix decomposes as A = I + (m/L)P where P is the cross-coset matching permutation. The eigenvalues are 1 ± m/L.
The connection to characters
The eigenvalue formula λ_j = (1/L) Σ R(ℓ) χ_j(ℓ) identifies the eigenvalues as inner products of the autocorrelation with additive characters of ℤ/Lℤ. For primitive-root primes, these characters are closely related to Dirichlet characters modulo p.
The paper
The Spectral Structure of Fractional Fields (PDF)
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