Fourier nonuniqueness sets for the hyperbola and the Perron-Frobenius operators
Let $Γ$ be a smooth curve or finite disjoint union of smooth curves in the plane and $Λ$ be any subset of the plane. Let $\mathcal X(Γ)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $Γ$ and are absolutely continuous with respect to the arc length measure on $Γ.$ Let $\mathcal{AC}(Γ,Λ)=\{μ\in \mathcal{X}(Γ) : \hatμ|_Λ=0\},$ then we prove the following results: \begin{enumerate}[(a)] \item For a rational perturbation of $Λ_β$ namely, $Λ_β^θ=\left((\mathbb Z+\{θ\})\times\{0\}\right)\cup\left(\{0\}\timesβ\mathbb Z\right),$ where $θ=1/{p},~\text{for some}~{p}\in\mathbb N,$ and $β$ is a positive real, $\mathcal{AC}\left(Γ,Λ_β^θ\right)$ is infinite-dimensional whenever $β>p.$ \smallskip \item For a rational perturbation of $Λ_γ$ namely, $Λ_γ^θ=\left((2\mathbb Z+\{2θ\})\times\{0\}\right)\cup\left(\{0\} \times2γ\mathbb Z\right),$ where $θ=1/q,~\text{for some}~q\in\mathbb N,$ and $γ$ is a positive real, $\mathcal{AC}\left(Γ_+,Λ_γ^θ\right)$ is infinite-dimensional whenever $γ>q.$ \end{enumerate}