On the Computational Problems of Upper Convex Densities for Self-Similar Sets with the Open Set Condition (Journal of Mathematical Study, 2024)
Let $E$ be a self-similar set satisfying the open set condition. Zhou and Fengposed an open problem in 2004 as follows: let $x ∈ E,$ under what conditions is therea set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) = \frac{\mathcal{H}^s(E \cap U_x)}{|U_x|^s}?$ The aim of thispaper is to present a solution of this problem. Under the assumption that there exists anonempty convex open set $V$ containing $E$ and satisfying the requirement of the openset condition, it is proved that if $x ∈ E$ and the upper convex density of $E$ at $x$ equals1, then there exists a convex set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) =\frac{\mathcal{H}^s(E∩U_x)}{|U_x|^s}.$ Finally, as an application of this result, an equivalent condition for $E_0 = E$ isgiven, where $E_0 =\{x∈E|\overline{D}^s_C(E,x)=1\}.$