  A pair $(\Gamma,\Lambda)$, where $\Gamma\subset\mathbb{R}^2$ is a locally
rectifiable curve and $\Lambda\subset\mathbb{R}^2$ is a {\em Heisenberg
uniqueness pair} if an absolutely continuous (with respect to arc length)
finite complex-valued Borel measure supported on $\Gamma$ whose Fourier
transform vanishes on $\Lambda$ necessarily is the zero measure. Recently, it
was shown by Hedenmalm and Montes that if $\Gamma$ is the hyperbola
$x_1x_2=M^2/(4\pi^2)$, where $M>0$ is the mass, and $\Lambda$ is the
lattice-cross $(\alpha\mathbb{Z}\times\{0\}) \cup
(\{0\}\times\beta\mathbb{Z})$, where $\alpha,\beta$ are positive reals, then
$(\Gamma,\Lambda)$ is a Heisenberg uniqueness pair if and only if $\alpha\beta
M^2\le4\pi^2$. 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 $L^\infty(\mathbb{R}_+)$ 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
$m,n\ge0$ spans a weak-star dense subspace of $H^\infty_+(\mathbb{R})$ if and
only if $0<\alpha\beta\le1$. 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.
