(untitled) (Journal of Mathematical Study, 2024)
Let E be a self-similar set satisfying the open set condition. Zhou and Feng posed an open problem in 2004 as follows: let x ∈ E, under what conditions is there a set U_x containing x with |U_x| > 0 such that D^s_C(E,x) = H^s(E U_x)|U_x|^s? The aim of this paper is to present a solution of this problem. Under the assumption that there exists a nonempty convex open set V containing E and satisfying the requirement of the open set condition, it is proved that if x ∈ E and the upper convex density of E at x equals 1, then there exists a convex set U_x containing x with |U_x| > 0 such that D^s_C(E,x) =H^s (E∩U_x)|U_x|^s. Finally, as an application of this result, an equivalent condition for E_0 = E is given, where E_0 ={x∈E|D^s_C(E,x)=1}.