On Gröchenig's Problem 4.12: Failure and Central-Dominance Repair of Beurling-Density Variation Diminution for Gaussian Shifts

This preprint presents an independent mathematical investigation motivated by Problem 4.12 in Karlheinz Gröchenig, Infinite totally positive matrices: some open problems, Acta Scientiarum Mathematicarum (2026), DOI: 10.1007/s44146-026-00264-3. Problem 4.12 asks whether, for a Pólya-frequency generator \(g\) and a bounded real coefficient sequence \(c=(c_k)_{k\in\mathbb Z}\), the lower and upper Beurling densities of the real zero set of \[ f=\sum_{k\in\mathbb Z} c_k\,g(\cdot-k) \] are bounded by the corresponding densities of coefficient sign changes. The present work studies the Gaussian generator \[ g(x)=e^{-\pi x^2}. \] It contains three principal results. Lower-density failure.A gliding-hump construction gives a bounded real coefficient sequence satisfying \[ D^-(c)=0, \qquad D^-(Z(f))\ge \frac12. \] Thus the same-direction lower-density inequality fails already for Gaussian shifts. Upper-density failure with a fixed periodic sign pattern.For every integer \(Q\ge1\), the manuscript constructs an absolutely summable, nowhere-zero coefficient sequence with \[ \operatorname{sgn}(c_k)=(-1)^{\lfloor k/Q\rfloor}, \] and hence \[ D^-(c)=D^+(c)=\frac1Q, \] while nevertheless \[ D^+(Z(f))=\infty. \] Thus even an exact periodic sign-change density does not control the upper density of real zeros. Central Gaussian Dominance.The paper gives an amplitude-sensitive sufficient condition under which coefficient sign changes and real zeros return to exact correspondence. If there exist strictly increasing control points \(x_k\to\pm\infty\) such that \[ |c_k|e^{-\pi(x_k-k)^2} > \sum_{j\ne k}|c_j|e^{-\pi(x_k-j)^2}, \] then each coefficient sign change corresponds to exactly one simple real zero, no additional real zeros occur, and the lower and upper densities are preserved. For the lattice choice \(x_k=k\), a sufficient condition is \[ \sup_k \sum_{j\ne k} \left|\frac{c_j}{c_k}\right| e^{-\pi(j-k)^2} <1. \] In particular, if \[ 0<m\le |c_k|\le M<\infty \] and \[ \frac{M}{m}<11.5694126702278\ldots, \] then \[ D^-(Z(f))=D^-(c), \qquad D^+(Z(f))=D^+(c). \] The manuscript therefore separates two distinct failure mechanisms—macroscopic spatial redistribution and amplitude compression—and identifies an amplitude-sensitive structural condition that blocks both. Cross-direction density inequalities are not addressed in this release, and no claim about their general status is made here. This version is released as a public preprint for mathematical inspection and independent verification. It has undergone extensive internal reconstruction and adversarial checking, but it is not peer reviewed and is not claimed to have been independently certified. No claim of worldwide novelty, priority over unidentified prior art, or formal verification is made. Original problem source:Karlheinz Gröchenig, Infinite totally positive matrices: some open problems, Acta Scientiarum Mathematicarum, 2026.DOI: 10.1007/s44146-026-00264-3 Version v2.0 DOI:10.5281/zenodo.23055369 Concept DOI / all versions:10.5281/zenodo.22661661 Earlier archival version:Version v1.0, DOI 10.5281/zenodo.22661662, was preserved as a restricted research state containing earlier review and development drafts.