Cartesian products of Sierpiński carpets do not attain their conformal dimension
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913051439005696 |
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| author | Anttila, Riku Eriksson-Bique, Sylvester |
| author_facet | Anttila, Riku Eriksson-Bique, Sylvester |
| contents | It is a long-standing open question to determine whether the Sierpiński 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 Sierpiński 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 Sierpiński carpet, the Sierpiński gasket, the Menger sponge, and the Laakso diamond. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19600 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cartesian products of Sierpiński carpets do not attain their conformal dimension Anttila, Riku Eriksson-Bique, Sylvester Metric Geometry 30L10, 20F65, 51F99, 53C23, 28A78 It is a long-standing open question to determine whether the Sierpiński 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 Sierpiński 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 Sierpiński carpet, the Sierpiński gasket, the Menger sponge, and the Laakso diamond. |
| title | Cartesian products of Sierpiński carpets do not attain their conformal dimension |
| topic | Metric Geometry 30L10, 20F65, 51F99, 53C23, 28A78 |
| url | https://arxiv.org/abs/2604.19600 |