Functions from Sobolev and Besov spaces with maximal Hausdorff dimension of the exceptional Lebesgue set

We prove that for $p>1$ and $0<\alpha<n/p$ there exists a function from the Bessel potentials class $J_\alpha(L^p(\mathbb R^n))$ such that the Hausdorff dimension of its exceptional Lebesgue set is $n-\alpha p$. We also show that such a function may be taken from the Besov class $B^\alpha_{p,q}(\mathbb R^n)$ with any $q>0$.