  We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in
(1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set
of full Hausdorff dimension. We apply Kova\v{c}'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 {\L}aba and Wang, we hence prove a
lack of valid relations between the Hausdorff dimension of a set and the range
of possible Fourier restriction exponents for measures supported in the set.
