The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps

A pair $(\mathrm{\Gamma },\mathrm{\Lambda })$, where $\mathrm{\Gamma }\subset {\mathbb{R}}^2$ is a locally rectifiable curve and $\mathrm{\Lambda }\subset {\mathbb{R}}^2$ is a {\em Heisenberg uniqueness pair} if an absolutely continuous (with respect to arc length) finite complex-valued Borel measure supported on $\mathrm{\Gamma }$ whose Fourier transform vanishes on $\mathrm{\Lambda }$ necessarily is the zero measure. Recently, it was shown by Hedenmalm and Montes that if $\mathrm{\Gamma }$ is the hyperbola $x_1{x}_2=M^2\mathrm{/}(4{\pi }^2)$, where $M>0$ is the mass, and $\mathrm{\Lambda }$ is the lattice-cross $(\alpha$\mathbb{Z}$\times{0}) \cup ({0}\times\beta$\mathbb{Z}$)$,where$\alpha,\beta$arepositivereals,then$(\Gamma,\Lambda)$isaHeisenberguniquenesspairifandonlyif$\alpha\beta M^2\le4\pi^2$. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so the theorem supplies very thin uniqueness sets for a class of solutions to this equation. The case of the semi-axis ${\mathbb{R}}_{+}$ as well as the holomorphic counterpart remained open. In this work, we completely solve these two problems. As for the semi-axis, we show that the restriction to ${\mathbb{R}}_{+}$ of the above exponential system spans a weak-star dense subspace of $L^{\mathrm{\infty }}({\mathbb{R}}_{+})$ if and only if $0<\alpha \beta <4$, based on dynamics of Gauss-type maps. This has an interpretation in terms of dynamical unique continuation. As for the holomorphic counterpart, we show that the above exponential system with $m,n\ge 0$ spans a weak-star dense subspace of $H_{+}^{\mathrm{\infty }}(\mathbb{R})$ if and only if $0<\alpha \beta \le 1$. To obtain this result, we need to develop new harmonic analysis tools for the dynamics of Gauss-type maps, related to the Hilbert transform. Some details are deferred to a separate publication.