On the Hausdorff measure of self-similar sets in Qpd

Abstract

The study of self-similar sets and their fractal dimensions has been a central topic in geometric measure theory and fractal geometry. In this paper, we extend classical results on self-similar sets to the p-adic settings. Specifically, we investigate the Hausdorff measure and dimension of self-similar sets in $Q_p^d$, the d-dimensional vector space over the p-adic numbers. Our main results provide necessary and sufficient conditions for the open set condition (OSC) and strong open set condition (SOSC) in $Q_p^d$ and establish the relationship between these conditions and the Hausdorff measure of the attractor.