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

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}