On fractals which are not so terrible (Q2773370)

The paper is concerned with studying topological and measure theoretical properties of NST sets. Let \(\Gamma\subset\mathbb R^n\) be a non-empty set such that either \(\text{Int}(\Gamma)=\Gamma\) or \(\text{Int}(\Gamma)=\emptyset\). The set \(\Gamma\) is NST (not so terrible), if there exists \(n\in\mathbb N\) having the property that for any dyadic cube \(Q\) with side-length \(l(Q)\leq 1\) and \(\text{Cl}(Q)\cap\partial\Gamma\neq\emptyset\), there is a dyadic cube \(P\subset Q\) having side-length \(l(P)=2^{-n}l(Q)\) such that \(\text{Int}(P)\cap\partial\Gamma=\emptyset\). Here \(\text{Int}(A)\), \(\text{Cl}(A)\), and \(\partial A\) are the interior, closure, and boundary of a set \(A\), respectively. NEWLINENEWLINENEWLINEIn the paper relations between porous and NST sets are considered. The author also proves that if the \(d\)-dimensional Hausdorff measure of a closed set scales like \(r^d\) in small balls having radius \(r\), then the set is NST. Finally, sufficient conditions, guaranteeing that elements of certain subfamilies of self-affine sets are NST, are given.