A pair (Î,Î), where ÎâR2 is a locally
rectifiable curve and ÎâR2 is a {\em Heisenberg
uniqueness pair} if an absolutely continuous (with respect to arc length)
finite complex-valued Borel measure supported on Î whose Fourier
transform vanishes on Î necessarily is the zero measure. Recently, it
was shown by Hedenmalm and Montes that if Î is the hyperbola
x1âx2â=M2/(4Ï2), where M>0 is the mass, and Î is the
lattice-cross (αZĂ{0})âȘ({0}ĂÎČZ), where α,ÎČ are positive reals, then
(Î,Î) is a Heisenberg uniqueness pair if and only if αÎČM2â€4Ï2. 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 R+â 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 R+â of the above exponential system
spans a weak-star dense subspace of Lâ(R+â) if and only if
0<αÎČ<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,nâ„0 spans a weak-star dense subspace of H+ââ(R) if and
only if 0<αÎČâ€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.Comment: 41 page