Polynomial Fourier decay holds for images of self-similar measures under sufficiently nonlinear real-analytic maps.
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The authors prove that the Fourier dimension Δ(s,t) of any (s,t)-Kakeya set in the plane satisfies 2st/(s+2t) ≤ Δ(s,t) ≤ min{s,2t} for 0<s,t<1, with analogous bounds in the Furstenberg and Fourier-direction variants.
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Fourier transform of nonlinear images of self-similar measures: qualitative aspects
Polynomial Fourier decay holds for images of self-similar measures under sufficiently nonlinear real-analytic maps.
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Fourier analytic variants of the Furstenberg and Kakeya problems
The authors prove that the Fourier dimension Δ(s,t) of any (s,t)-Kakeya set in the plane satisfies 2st/(s+2t) ≤ Δ(s,t) ≤ min{s,2t} for 0<s,t<1, with analogous bounds in the Furstenberg and Fourier-direction variants.