Fourier nonuniqueness sets for the hyperbola and the Perron-Frobenius operators (2009.09516v1)

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}