LARGE SETS WITHOUT FOURIER RESTRICTION THEOREMS

We construct a function that lies in L-p(R-d) for every p is an element of (1, infinity] and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovac'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 Laba 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.