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.
A measure with small support and p-summable Fourier transform
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We construct a probability measure $\mu$ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu}\in L_{p}(\mathbb{R}^d)$.
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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.