An equivalent condition for the self similar sets on the real line to have best coverings (Q383615)

Let \(E\) be the self similar set on \(I=[0,1]\) for iterated function system (IFS) \(\{S_1,\dots,S_m\}\), \(0<c_i<1\) be the similarity ratio of \(S_i\), \(i=1,\dots,m\). Let \(x\in E\), denote \(\overline{D^s_C}(E,x)\) to be the upper convex density of \(E\) at \(x\) and \(E_0=\{x\in E:\;\overline{D^s_C}(E,x)=1\}\). The authors proved that if the IFS satisfy the open set condition, then \(E\) has a best covering if and only if \(E_0=E\) (i.e. an equivalent condition). The authors claimed that this result partially gives answer to one conjecture proposed by \textit{Z. Zhou} and \textit{L. Feng} [Nonlinearity 17, No. 2, 493--502 (2004; Zbl 1047.28002)].