Large sets without Fourier restriction theorems

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