For every q=αγ>1 and every pair of shifts, the shifted lattice cross on the positive hyperbola branch has an infinite-dimensional Fourier pre-annihilator: uniqueness fails.
Hennion,Sur un théorème spectral et son application aux noyaux lipchitziens, Proc
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Heisenberg Uniqueness Pairs for a Hyperbola Branch: Supercritical Nonuniqueness for Shifted Lattice Crosses
For every q=αγ>1 and every pair of shifts, the shifted lattice cross on the positive hyperbola branch has an infinite-dimensional Fourier pre-annihilator: uniqueness fails.