  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}
