Spectral Crowding and Compression Limits for Arithmetic Covariance Operators

Abstract

We prove quantitative spectral crowding for a complete arithmetic covariance operator in the inverse metric of its actual positive receiver. A growing space of compactly supported oscillatory profiles gives linearly many eigenvalues near one, including after removal of an expanding frequency interval. The construction controls the entire complex coefficient space, its exact polar correction, and its full Fourier leakage. A logarithmic archimedean energy estimate sharpens the admissible frequency range. We then transfer quantitative prime-number-theorem errors through the complete arithmetic smoothing and obtain explicit lower bounds on the ranks of positive source compressions whose complete residuals vanish. The Vinogradov–Korobov error gives a stretched-exponential rank requirement. Using the seven-eighths zero-free theorem and the corresponding quantitative prime-number-theorem bound, we obtain a rank lower bound of Ω(M^(1/24)). Classical min–max and approximation theory turn the arithmetic spectral estimate into best-rank and effective-dimension consequences. The arithmetic estimate retains the original weighted domain and the complete source. The rank conclusion applies along any unbounded cutoff sequence with vanishing operator-norm residual, and its positive lower constant may depend on that sequence.