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arXiv · 2604.19600

Cartesian products of Sierpi\'nski carpets do not attain their conformal dimension

Riku Anttila·Sylvester Eriksson-Bique
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Abstract

It is a long-standing open question to determine whether the Sierpi\'nski carpet attains its conformal dimension or not. While this problem remains unresolved, we prove that Cartesian products $\mathbb{S}^k$, where $\mathbb{S}$ is the Sierpi\'nski carpet and $k \geq 2$, do not attain their conformal dimension. Our approach is based on the Sobolev spaces and energy measures on $\mathbb{S}$ -- constructed by Shimizu, Kigami, and Murugan and Shimizu -- together with a certain singularity result of energy measures from the theory of analysis on fractals. This work formulates a general non-attainment result of conformal dimension for product metric spaces $X^k$ for $k \geq 2$ in terms of self-similarity and energy measures of the factor $X$. It applies, in particular, to the cases where $X$ is the Sierpi\'nski carpet, the Sierpi\'nski gasket, the Menger sponge, and the Laakso diamond.

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Riku Anttila, Sylvester Eriksson-Bique. 2026-04-21. Cartesian products of Sierpi\'nski carpets do not attain their conformal dimension. https://arxiv.org/abs/2604.19600

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