Beurling Nyman Geometry and Gram Matrix Structure, Ladder Density and Polynomial Decay via Mellin Smoothing
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| Format: | Preprint |
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2025
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| _version_ | 1866915567505506304 |
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| author | Carvill, Hugh |
| author_facet | Carvill, Hugh |
| contents | We study the Beurling Nyman (BN) family $f_θ(x) = \{θ/x\} - θ\{1/x\}$ in $L^2((0,1])$ through a multiscale ladder parameterisation $θ_{j,k} = 2^{-j}3^{-k}$ 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 $$|\langle g_{θ_{j,k}}, g_{θ_{j',k'}} \rangle| \ll_m \bigl(1 + c d((j,k),(j',k'))\bigr)^{-m}$$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_18132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beurling Nyman Geometry and Gram Matrix Structure, Ladder Density and Polynomial Decay via Mellin Smoothing Carvill, Hugh Classical Analysis and ODEs Number Theory 11M26, 42C10, 46E22 We study the Beurling Nyman (BN) family $f_θ(x) = \{θ/x\} - θ\{1/x\}$ in $L^2((0,1])$ through a multiscale ladder parameterisation $θ_{j,k} = 2^{-j}3^{-k}$ 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 $$|\langle g_{θ_{j,k}}, g_{θ_{j',k'}} \rangle| \ll_m \bigl(1 + c d((j,k),(j',k'))\bigr)^{-m}$$ 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. |
| title | Beurling Nyman Geometry and Gram Matrix Structure, Ladder Density and Polynomial Decay via Mellin Smoothing |
| topic | Classical Analysis and ODEs Number Theory 11M26, 42C10, 46E22 |
| url | https://arxiv.org/abs/2510.18132 |