The Entropy and Hausdorff Dimension of self-similar sets

TL;DR

This paper proves that for any self-similar set in a unit cube, its Hausdorff dimension equals its entropy, confirming a conjecture and deepening understanding of fractal geometry and information theory.

Contribution

The paper establishes the equality of Hausdorff dimension and entropy for self-similar sets, resolving a conjecture by Adamczewski and Bell.

Findings

Hausdorff dimension equals entropy for self-similar sets

Confirmed a conjecture in fractal geometry

Provides a new link between geometric and informational properties

Abstract

Given a $k$-self similar set $X\subset [0,1]^{d}$ we calculate both its Hausdorff dimension and its entropy, and show that these two quantities are in fact equal. This affirmatively resolves a conjecture of Adamczewski and Bell.