**Silvano Delladio ^{[1]}**

[1] : Department of Mathematics, Università degli Studi di Trento, via Sommarive 14, Povo, Trento, Italy

**Abstract:**
Given \(m>0\) and a measurable set \(E\subset\mathbb{R}^n\), \(E^{(m)}\) denotes the set of \(m\)-density points of \(E\), namely the points \(x\in\mathbb{R}^n\) at which \({\cal L}^n(B(x,r)\backslash E)\) is an
infinitesimal of order greater than \(r^{m}\) (as \(r\to 0\)). We investigate the size of \(E^{(m)}\) in the particular case when \(E\) is a generalized Cantor set in \(\mathbb{R}\).
Moreover we prove the following result. Let \(\varphi\in C^h(\Omega)\) and \(\Phi\in C^h(\Omega;\mathbb{R}^n))\), where \(\Omega\) is an open subset
of \(\mathbb{R}^n\) and \(h\geq 1\). If \(K:=\{ x\in\Omega\,\vert\,\nabla\varphi(x)=\Phi(x)\}\) then the graph of \(\varphi_{\vert\Omega\cap K^{(n+h)}}\) is a \(n\)-dimensional \(C^{h+1}\)-rectifiable set.

**References**

[1] G. Alberti, *A Lusin type theorem for gradients*, J. Funct. Anal 100 (1991),110-118.
[MR1124295]

[2] G. Anzellotti and R. Serapioni, *\(\scr C_k\)-rectifiable sets*, J. Reine Angew. Math. 453 (1994), 1-20.
[MR1285779]

[3] A.-P. Calderón and A. Zygmund, *Local properties of solutions of elliptic partial differential equations*, Studia Math. 20 (1961), 171-225.
[MR0136849]

[4] S. Delladio, *Dilatations of graphs and Taylor's formula: some results about convergence*, Real Anal. Exchange 29 (2003/04), no. 2, 687-712.
[MR2083806]

[5] S. Delladio, *Functions of class \(C^1\) subject to a Legendre condition in an enhanced density set *, Rev. Mat. Iberoam. 28 (2012), no. 1, 127-140.
[MR2904134]

[6] H. Federer, *Geometric measure theory*, Springer-Verlag, New York 1969.
[MR0257325]

[7] P. Mattila, *Geometry of sets and measures in Euclidean spaces*, Cambridge University Press, Cambridge 1995.
[MR1333890]

[8] W. P. Ziemer, *Weakly differentiable functions*, Grad. Texts in Math., 120, Springer-Verlag, New York 1989.
[MR1014685]

Home Riv.Mat.Univ.Parma