If a function lies in L^p and its Fourier transform is supported on a small fractal set on the moment curve, then for p up to a sharp threshold (d^2+d+2α)/(2α) for d≥3, and 4/α for d=2, the function is identically zero.
Demeter.Fourier restriction, decoupling, and applications, volume 184 ofCambridge Studies in Advanced Mathematics
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$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve
If a function lies in L^p and its Fourier transform is supported on a small fractal set on the moment curve, then for p up to a sharp threshold (d^2+d+2α)/(2α) for d≥3, and 4/α for d=2, the function is identically zero.