Sets of Salem type and sharpness of the $L^2$-Fourier restriction theorem

Abstract

We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the $L^2$-Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.

21 pages, revised version, to appear in Trans. Amer. Math. Soc

References in corpus (1)

• On the sharpness of Mockenhaupt's restriction theorem

Cited by in corpus (1)

• Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Restriction Theorem in Higher Dimensions