Heisenberg uniqueness pairs for the finitely many parallel lines with an irregular gap (Q2124655)

\textit{H. Hedenmalm} and \textit{A. Montes-Rodríguez} [Ann. Math. (2) 173, No. 3, 1507--1527 (2011; Zbl 1227.42002)] introduced the notion of the Heisenberg uniqueness pair. The Heisenberg uniqueness pair, up to some extent, is similar to the annihilating pair of Borel measurable sets of positive measure as studied by \textit{V. Havin} and \textit{B. Jöricke} [The uncertainty principle in harmonic analysis. Berlin: Springer-Verlag (1994; Zbl 0827.42001)]. Let \(\Gamma\) be a finite disjoint union of smooth curves in \(\mathbb{R}^2\). Let \(X(\Gamma)\) be the space of all finite complex-valued Borel measures \(\mu\) in \(\mathbb{R}^2\) which are supported on \(\Gamma\) and absolutely continuous with respect to the arc length measure on \(\Gamma\). For \((\xi, \eta)\in\mathbb{R}^2\), the Fourier transform of \(\mu\) is defined by \(\hat{\mu}(\xi, \eta)=\int_\Gamma e^{i\pi(x\xi+y\eta)}\, d\mu(x,y)\). Let \(\Lambda\) be a set in \(\mathbb{R}^2\). The pair \((\Gamma,\Lambda)\) is called a Heisenberg uniqueness pair for \(X(\Gamma)\) if any \(\mu\in X(\Gamma)\) satisfying \(\hat{\mu}|_{\Lambda}=0\), implies \(\mu = 0\). In this paper, the authors give necessary and sufficient conditions for \(\Lambda\subseteq\mathbb{R}^2\) such that any finite Borel measure which is supported on a certain system of finitely many parallel lines with an irregular gap and absolutely continuous with respect to the arc length measure on \(\Gamma\) and whose Fourier transform vanishes on \(\lambda\), is the zero measure. They observe that the size of determining sets \(\Lambda\) for \(X(\Gamma)\) depends on the choice of several lines as well as their irregular distribution that further relates to a phenomenon of the interlacing of certain trigonometric polynomials.