Packing dimensions of the divergence points of self-similar measures with OSC

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 \(\frac{\log \mu (B(x,r))}{\log r}\) as \(r\searrow 0.\) Our main result solves the conjecture about packing dimension posed by Olsen and Winte (J London Math Soc, 67(2), pp 103–122, 2003) and generalizes the result in (Adv Math, 214, pp 267–287, (2007)).