We show that if E is an s-regular set in R^2 for which the triple integral \int_E \int_E \int_E c(x, y, z)^{2s} dH^sx dH^sy dH^sz of the Menger curvature c is finite and if 0 < s ≤ 1/2, then H^s almost all of E can be covered with countably many C^1 curves. We give an example to show that this is false for 1/2 <s< 1.

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We consider random Schrödinger equations on$R$^{$d$}for$d$≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$ . The space and time variables scale as $ x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} $ . We prove that, in the limit λ → 0, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable$x$for arbitrary$L$^{2}initial data.