Geodesic
Functions from Sobolev and Besov spaces with maximal Hausdorff dimension of the exceptional Lebesgue set
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 5, pp. 145-153

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We prove that for $p>1$ and $0\alpha$ 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 $q>0$. 
@article{FPM_2013_18_5_a7,
     author = {V. G. Krotov and M. A. Prokhorovich},
     title = {Functions from {Sobolev} and {Besov} spaces with maximal {Hausdorff} dimension of the exceptional {Lebesgue} set},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {145--153},
     publisher = {mathdoc},
     volume = {18},
     number = {5},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2013_18_5_a7/}
}
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V. G. Krotov; M. A. Prokhorovich. Functions from Sobolev and Besov spaces with maximal Hausdorff dimension of the exceptional Lebesgue set. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 5, pp. 145-153. http://geodesic.mathdoc.fr/item/FPM_2013_18_5_a7/