Large sets without Fourier restriction theorems

Fourier restriction inequalities enable the restriction of the Fourier transform to suitable sets of null Lebesgue measure. Fourier restriction to submanifolds is closely related to problems in PDE and in this talk we mainly focus on restrictions to fractals. The class of fractals is extensive and many different Fourier restriction properties can be observed, but necessary conditions are not well understood. In this direction, we show that all Fourier restriction sets avoid a universal set of full Hausdorff dimension.