Packing dimensions of the divergence points of self-similar measures with the open set condition

Abstract

Let $\mu$ be the self-similar measure supported on the self-similar set $K$ with open set condition. In this article, we discuss the packing dimension of the set $\{x\in K:A(\frac{\log \mu (B(x,r))}{\log r})=I\}$ for $I\subseteq \mathbb{R}$, where $A(\frac{\log \mu (B(x,r))}{\log r})$ denotes the set of accumulation points of \frac{\logμ(B(x,r))}{\log r}$as$r\searrow0$. Our main result solves the conjecture about packing dimension posed by Olsen and Winter \cite{OlsWin} and generalizes the result in \cite{BaeOlsSni}.

13 pages