Random Cantor sets of dimension d < 2/3 in R satisfy the fractal uncertainty principle with exponent >= 1/2 - 3d/4 - with overwhelming probability, via Fourier decay from concentration of measure.
Cohen, Fractal uncertainty for discrete 2D Cantor sets
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Fractal uncertainty principle for random Cantor sets
Random Cantor sets of dimension d < 2/3 in R satisfy the fractal uncertainty principle with exponent >= 1/2 - 3d/4 - with overwhelming probability, via Fourier decay from concentration of measure.