Large sets without Fourier restriction theorems

Abstract

We construct a function that lies in L p ( R d ) L^p(\mathbb {R}^d) for every p ∈ ( 1 , ∞ ] p \in (1,\infty ] and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač’s maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set.