Fourier nonuniqueness sets for the hyperbola and the Perron-Frobenius operators

TL;DR

This paper investigates the conditions under which Fourier nonuniqueness sets for measures supported on smooth curves in the plane are infinite-dimensional, focusing on specific rational perturbations of lattice-like sets and their impact on the Fourier transform.

Contribution

The paper establishes new results on the infinite-dimensionality of measure spaces with vanishing Fourier transform on certain rationally perturbed lattice sets, extending understanding of Fourier nonuniqueness.

Findings

Infinite-dimensionality for rational perturbations with parameters exceeding certain thresholds.

Explicit construction of nontrivial measures supported on curves with zero Fourier transform on perturbed sets.

Extension of nonuniqueness results to specific geometric configurations involving lattices and curves.

Abstract

Let $\Gamma$ be a smooth curve or finite disjoint union of smooth curves in the plane and $\Lambda$ be any subset of the plane. Let $\mathcal X(\Gamma)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $\Gamma$ and are absolutely continuous with respect to the arc length measure on $\Gamma.$ Let $\mathcal{AC}(\Gamma,\Lambda)=\{\mu\in \mathcal{X}(\Gamma) : \hat\mu|_{\Lambda}=0\},$ then we prove the following results: \begin{enumerate}[(a)] \item For a rational perturbation of $\Lambda_\beta$ namely, $\Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right)\cup\left(\{0\}\times\beta\mathbb Z\right),$ where $\theta=1/{p},~\text{for some}~{p}\in\mathbb N,$ and $\beta$ is a positive real, $\mathcal{AC}\left(\Gamma,\Lambda_\beta^\theta\right)$ is infinite-dimensional whenever $\beta>p.$ \smallskip \item For a rational perturbation of $\Lambda_\gamma$ namely, $\Lambda_\gamma^\theta=\left((2\mathbb Z+\{2\theta\})\times\{0\}\right)\cup\left(\{0\} \times2\gamma\mathbb Z\right),$ where $\theta=1/q,~\text{for some}~q\in\mathbb N,$ and $\gamma$ is a positive real, $\mathcal{AC}\left(\Gamma_+,\Lambda_\gamma^\theta\right)$ is infinite-dimensional whenever $\gamma>q.$ \end{enumerate}