Every Kakeya set in F_q^d supports a probability measure with Fourier transform bounded by q^{-1} at nonzero frequencies, and this bound is sharp in all dimensions d at least 2.
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Fourier analytic properties of Kakeya sets in finite fields
Every Kakeya set in F_q^d supports a probability measure with Fourier transform bounded by q^{-1} at nonzero frequencies, and this bound is sharp in all dimensions d at least 2.