The notion of Heisenberg uniqueness pair has been introduced by Hedenmalm
and Montes-Rodriguez (Ann. of Math. 2011) as a version of the uncertainty
principle, that is, a nonzero function and its Fourier transform both
cannot be too small simultaneously. 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 $\mathcal{AC}(\Gamma,\Lambda)=\{\mu\in
\mathcal{X}(\Gamma) : \widehat\mu|_{\Lambda}=0\},$ then we say that
$\Lambda$ is a *Fourier uniqueness set* for $\Gamma$ or
$(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair, if
$\mathcal{AC}(\Gamma,\Lambda)={0}.$

In this talk, we will discuss the following: Let $\Gamma$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $\Lambda_\beta^\theta$ be the lattice-cross defined by \begin{equation} \Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right) \cup \left(\{0\}\times\beta\mathbb Z\right), \end{equation} where $\beta$ is a positive real and $\theta=1/{p}$, for some $p\in\mathbb N,$ then $\left(\Gamma,\Lambda_\beta^\theta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq{p}.$ Moreover, the space $\mathcal{AC}\left(\Gamma,\Lambda_\beta^\theta\right)$ is infinite-dimensional provided $\beta>p.$

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Last updated: 22 Jun 2024