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Hausdorff dimension of a class of two-dimensional self-similar sets | Synapse

March 13, 2026

Hausdorff dimension of a class of two-dimensional self-similar sets

Key Points

The research aims to explore the Hausdorff dimension of two-dimensional self-similar sets with overlapping copies.
Analyzed a specific class of two-dimensional homogeneous self-similar sets.
Utilized the configuration concept to investigate properties of these sets.
Formulated and solved an equation to find the largest positive root.
Proved that the Hausdorff dimension can be determined under the specified conditions.
Identified the relationship between parameters leading to the Hausdorff dimension.
Established that overlaps affect the dimensions of self-similar sets.

Abstract

Let Formula: see text be a real number, we consider a class Formula: see text of two-dimensional homogeneous self-similar sets Formula: see text such that a pair of copies overlap. Based on the configuration concept by 1, we prove that Formula: see text, where Formula: see text is the largest positive root of the equation Formula: see text for some positive integers Formula: see text with Formula: see text.

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Cite This Study

Shaza Nammoura (Fri,) studied this question.

synapsesocial.com/papers/69b3ab8002a1e69014ccc6ed https://doi.org/https://doi.org/10.1142/s0218348x26500659