Singular sets of Sobolev functions

We are interested in finding Sobolev functions with “large” singular sets. Given $\text{N,k∈}\text{N}$, 1<p<∞, kp<N, for any compact subset A of $\text{R}{}^{\mathrm{N}}$, such that its upper box dimension is less than N?kp, we construct a Sobolev function $\text{u∈}\text{W}{}^{\mathrm{k,p}}\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N?kp. To cite this article: D. ?ubrini?, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.