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

TL;DR

This paper determines the packing dimension of divergence points of self-similar measures with the open set condition, confirming a conjecture and extending previous results in fractal geometry.

Contribution

It proves a conjecture about the packing dimension of divergence points for self-similar measures, generalizing earlier work.

Findings

Solved the conjecture on packing dimension of divergence points

Generalized previous results in the literature

Provided a comprehensive analysis under 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{\log\mu(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}.