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In this paper we investigate the restriction problem. More precisely, we givesufficient conditions for the failure of a set $E$ in $\mathbb{R}^n$ to havethe $p$-restriction property. We also extend the concept of spectral synthesisto $L^p(\mathbb{R}^n)$ for sets of $p$-restriction when $p > 1$. We use ourresults to show that there are $p$-values for which the unit sphere is a set of$p$-spectral synthesis in $\mathbb{R}^n$ when $n \geq 3$.

Sets of $p$-restriction and $p$-spectral synthesis