Algorithms to test open set condition for self-similar set related to P.V. numbers

TL;DR

This paper develops an efficient algorithm to determine whether certain self-similar sets, defined by P.V. numbers and specific parameters, satisfy the open set condition, which is important for understanding their geometric structure.

Contribution

The paper introduces a novel algorithm for testing the open set condition in self-similar sets associated with P.V. numbers, advancing computational methods in fractal geometry.

Findings

Algorithm efficiently tests open set condition

Applicable to self-similar sets with P.V. number parameters

Enhances understanding of fractal set separations

Abstract

Fix a P.V. number $\lambda ^{-1}>1.$ Given $\mathbf{p}=(p_{1},\cdots,p_{m})\in \mathbb{N}^{m}$, $\mathbf{b}=(b_{1},\cdots,b_{m})\in \mathbb{Q^{m}$, for the self-similar set $E_{\mathbf{p},\mathbf{b}}=\cup_{i=1}^{m}(\lambda ^{p_{i}}E_{\mathbf{p},\mathbf{b}}+b_{i})$ we find an efficient algorithm to test whether $E_{\mathbf{p},\mathbf{b}}$ satisfies the open set condition (strong separation condition) or not.