For every 2<p<∞, a probability measure on R^d exists with \hat μ ∈ L^p supported on a compact set of zero H^{2d/p} measure.
Beurling, On a closure problem, Ark
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A measure with small support and p-summable Fourier transform
For every 2<p<∞, a probability measure on R^d exists with \hat μ ∈ L^p supported on a compact set of zero H^{2d/p} measure.