  We obtain new Fourier interpolation and -uniqueness results in all
dimensions, extending methods and results by the first author and M. Sousa, and
by the second author. We show that the only Schwartz function which, together
with its Fourier transform, vanishes on surfaces close to the origin-centered
spheres whose radius are square roots of integers, is the zero function. In the
radial case, these surfaces are spheres with perturbed radii, while in the
non-radial case, they can be graphs of continuous functions over the sphere. As
an application, we translate our perturbed Fourier uniqueness results to
perturbed Heisenberg uniqueness for the hyperbola, using the interrelation
between these fields introduced and studied by Bakan, Hedenmalm,
Montes-Rodriguez, Radchenko and Viazovska.
