Develops a quantitative flatness framework to obstruct Fourier restriction, L^p-improving, and Fourier decay estimates for measures, applied to bound Fourier dimensions of surfaces, curves, Patterson-Sullivan measures, and self-affine sets.
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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.
Borel sets with Fourier dimension at least 2 have distance sets of full Hausdorff dimension in any ambient dimension d, and sets with Fourier spectrum at least d/4 + 1 at theta = 1/2 also achieve this even when their Fourier dimension is zero provided d is at least 4.
Provides zero-full laws for measures of approximation sets and exact Fourier dimensions, showing non-Salem property except in 1D and that product Fourier dimension equals the minimum of the two.
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Quantitative flatness and obstructions in Fourier analysis
Develops a quantitative flatness framework to obstruct Fourier restriction, L^p-improving, and Fourier decay estimates for measures, applied to bound Fourier dimensions of surfaces, curves, Patterson-Sullivan measures, and self-affine sets.
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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.
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On Fourier decay and the distance set problem
Borel sets with Fourier dimension at least 2 have distance sets of full Hausdorff dimension in any ambient dimension d, and sets with Fourier spectrum at least d/4 + 1 at theta = 1/2 also achieve this even when their Fourier dimension is zero provided d is at least 4.
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Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation
Provides zero-full laws for measures of approximation sets and exact Fourier dimensions, showing non-Salem property except in 1D and that product Fourier dimension equals the minimum of the two.