Dimension Of Inhomogeneous Sub-Self-Similar Sets
Shivam Dubey, Saurabh Verma
TL;DR
This work introduces inhomogeneous sub-self-similar (ISSS) sets, extending Falconer’s sub-self-similar framework by incorporating a condensation set $C$ and studying how ISSS dimensions relate to the Hausdorff and box dimensions of $C$ and a homogeneous SSS base. The authors provide concrete construction methods for ISSS sets, including a canonical form $F=E\cup O_S$ derived from a sub-SSS $E$ with $\Theta(S)=E$, and they derive sharp dimension bounds using the $\,\delta$-covering technique and Fraser’s stopping-set framework. Key results include $\dim_H(E\cup O_S)=\max\{s,\dim_H C\}$ and upper/lower box-dimension estimates with explicit dependence on contraction ratios and covering regularity, along with a continuity theorem for $\dim_H$ via IOSC-approximations. They also develop a product-IFS theory showing that inhomogeneous products preserve separation properties and yield inhomogeneous-type product measures. Overall, the paper advances dimension theory for ISSS fractals and opens avenues for further analysis of dimension spectra and higher-complexity constructions.
Abstract
In this paper, we introduce the concept of Inhomogeneous sub-self-similar (ISSS) sets, building upon the foundations laid by Falconer (Trans. Amer. Math. Soc. 347 (1995) 3121-3129) in the study of sub-self-similar sets and drawing inspiration from Barnsley's work on inhomogeneous self-similar sets (Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 24). We explore a range of examples of ISSS sets and elucidate a method to construct ISSS sets. We also investigate the upper and lower box dimensions of ISSS sets and discuss the continuity of the Hausdorff dimension.