Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificates

Abstract

We approximate the complete local Nyman–Beurling projection problem uniformly before optimizing its Dirichlet-polynomial coefficients. A support-sensitive tail estimate retains the shared arithmetic head, the height-dependent mixed Bernoulli covariances, both exact coefficient constraints and the entire norm and mean errors. Its error depends on the separation of the support's divisor frequencies. Dense supports give the regime N=o(T); finite unions of fixed geometric lattices admit an O(T^(-1)) optimized-gain error even when N=exp(T^β) for any fixed β>0. Model maximizers then have an explicit bound on their loss in true gain. In contrast, the best true gain over each fixed geometric module, allowing arbitrary degrees and height-dependent coefficients, becomes smaller than every positive threshold on positive-lower-density sets of heights. Joint Hardy-kernel projection also bounds the complex defect required to escape such modules. Thus accurate finite models and adequate approximation dictionaries are separate requirements. The proof combines classical separated-frequency estimates with varying primitive amplitudes, an exact cotangent tail form and a compensated arithmetic mean.