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

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 $\mathcal{A}\mathcal{C}(\Gamma ,\Lambda )=\{\mu \in \mathcal{X}(\Gamma ):\hat{\mu }{|}_{\Lambda }=0\},$ 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, $\mathcal{A}\mathcal{C}\left(\Gamma ,{\Lambda }_{\beta }^{\theta }\right)$ is infinite-dimensional whenever $\beta >p.$ \smallskip \item For a rational perturbation of ${\Lambda }_{\gamma }$ namely, ${\Lambda }_{\gamma }^{\theta }=\left((2\mathbb{Z}+\{2\theta \})\times \{0\}\right)\cup \left(\{0\}\times 2\gamma \mathbb{Z}\right),$ where $\theta =1/q,\text{for some}q\in \mathbb{N},$ and $\gamma$ is a positive real, $\mathcal{A}\mathcal{C}\left({\Gamma }_{+},{\Lambda }_{\gamma }^{\theta }\right)$ is infinite-dimensional whenever $\gamma >q.$ \end{enumerate}

20 pages