Partial Resolutions of the Shifted Laguerre Converse: Gaussian Drift and Directional Density Barriers

Abstract

Let $f$ be a real entire function in the strip class $\mathcal{S}(A)$, and define $f_\mu(x) = f(x + i\mu) + f(x - i\mu)$. Csordas and Escassut conjectured that the strict shifted Laguerre condition $L[f_\mu](x) > 0$ for every real $x$ and every nonzero $\mu$ forces $f$ to belong to the Laguerre–Pólya class $\mathcal{LP}$. We establish several partial results toward this conjecture: Gaussian Case: We prove the conjecture when the Gaussian factor in the canonical product is nontrivial. Directional Density Obstruction: In the Gaussian-free case, we derive a strict directional zero-density obstruction: any finite-density direction carrying a nonreal directed escape must satisfy $$A \cdot \overline{D}_\sigma > 1.0138$$ For Cartwright functions, this implies that $\tau A \le 1.0138\pi$ forces $f$ to belong to the Laguerre–Pólya class. Height-Sensitive Refinement: We obtain a refinement in which $A$ is replaced by $\sqrt{A^2 - \eta^2}$ when the escape has limiting height $\eta$. Structural Obstructions: The argument combines the geometry of the level set $\operatorname{Re} f = 0$, shadow-disk coverings, sparse estimates for canonical products, and a real-axis energy identity for the logarithmic derivative. Model Class Counterexamples: Finally, we rule out a broad model class of counterexamples: if, up to finitely many exceptions, the zeros in the upper half-plane form a finite union of horizontal lattices at a common height, then the shifted Laguerre condition cannot hold. This obstruction follows from a Poisson-kernel representation and the Fourier–Bohr spectrum of an almost-periodic boundary field. Together, these results impose explicit geometric and density constraints that any counterexample to the full conjecture would have to evade.