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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.12054 |
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| _version_ | 1866912587788058624 |
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| author | Skvortsov, Valentin |
| author_facet | Skvortsov, Valentin |
| contents | We introduce an analogue of Riesz $s$-potetial and $s$-energy, $0<s<1$, of a mass distribution $μ$ on the Cantor dyadic group $G$ by defining a respactive $s$-kernel. Then we relate Hausdorff dimension of a set $E\subset G$ to the value of $s$-energy of the mass distribution $μ$ on this set $E$. Namely we prove that if on a set $E$ there exists a mass distribution $μ$ with finite $s$-energy, then the Hausdorff dimension of $E$ is at least $s$.
The same condiion can be expressed also in terms of Fourier coefficients of $μ$ with respect to Walsh system on the group $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analogues of $s$-potential and $s$-energy of mass distribution on Cantor dyadic group and their relation to Hausdorff dimension Skvortsov, Valentin Functional Analysis 42C40 (Primary), 43A75. 28A78 (Secondary) We introduce an analogue of Riesz $s$-potetial and $s$-energy, $0<s<1$, of a mass distribution $μ$ on the Cantor dyadic group $G$ by defining a respactive $s$-kernel. Then we relate Hausdorff dimension of a set $E\subset G$ to the value of $s$-energy of the mass distribution $μ$ on this set $E$. Namely we prove that if on a set $E$ there exists a mass distribution $μ$ with finite $s$-energy, then the Hausdorff dimension of $E$ is at least $s$. The same condiion can be expressed also in terms of Fourier coefficients of $μ$ with respect to Walsh system on the group $G$. |
| title | Analogues of $s$-potential and $s$-energy of mass distribution on Cantor dyadic group and their relation to Hausdorff dimension |
| topic | Functional Analysis 42C40 (Primary), 43A75. 28A78 (Secondary) |
| url | https://arxiv.org/abs/2509.12054 |