Beurling Nyman Geometry and Gram Matrix Structure, Ladder Density and Polynomial Decay via Mellin Smoothing

Abstract

We study the Beurling Nyman (BN) family $f_\theta(x) = \{\theta/x\} - \theta\{1/x\}$ in $L^2((0,1])$ through a multiscale ladder parameterisation $\theta_{j,k} = 2^{-j}3^{-k}$ and the associated Gram matrix structure indexed by ladder distance. Using Mellin analysis and a controlled smoothing operator, we establish a rigorous polynomial decay envelope for off-diagonal Gram entries. Specifically, for a Gaussian-type Mellin multiplier we prove that $$|\langle g_{\theta_{j,k}}, g_{\theta_{j',k'}} \rangle| \ll_m \bigl(1 + c d((j,k),(j',k'))\bigr)^{-m}$$ for any $m$ in $\mathbb{N}$, where $c = \min\{\log 2, \log 3\}$ and $d$ denotes the ladder distance. As a consequence we obtain block-compressibility of Gram rows for $m > 2$. These results provide a rigorous foundation for sparsity phenomena in the BN system and support constructive spectral approaches. The analysis is unconditional and independent of any hypothesis concerning the zeros of the zeta function.