Power Laws for the Favard Length Problem in $\mathbb{R}^d$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914018669625344 |
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| author | Marshall, Caleb |
| author_facet | Marshall, Caleb |
| contents | We prove a power law for the asymptotic decay of the Favard length of neighbourhoods of certain self-similar sets in $\mathbb{R}^d$ with $d \geq 2$. These self-similar sets are generalizations of the so-called four-corner Cantor set to higher dimensions, as well as to a more general class of rational digit sets. When $d \geq 3$, our estimates are the first such non-trivial asymptotic upper bounds for the Favard length problem. The extension to a new class of digit sets (which is new even when $d = 2$, but holds for $d \geq 2$ generally) uses the work of G. Kiss, I. Laba, G. Somlai and the author on vanishing sums of roots of unity and divisibility by many cyclotomic polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power Laws for the Favard Length Problem in $\mathbb{R}^d$ Marshall, Caleb Classical Analysis and ODEs Metric Geometry Number Theory 28A78, 28A80 (primary), 11C08, 42B05, 11B75 (secondary) We prove a power law for the asymptotic decay of the Favard length of neighbourhoods of certain self-similar sets in $\mathbb{R}^d$ with $d \geq 2$. These self-similar sets are generalizations of the so-called four-corner Cantor set to higher dimensions, as well as to a more general class of rational digit sets. When $d \geq 3$, our estimates are the first such non-trivial asymptotic upper bounds for the Favard length problem. The extension to a new class of digit sets (which is new even when $d = 2$, but holds for $d \geq 2$ generally) uses the work of G. Kiss, I. Laba, G. Somlai and the author on vanishing sums of roots of unity and divisibility by many cyclotomic polynomials. |
| title | Power Laws for the Favard Length Problem in $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs Metric Geometry Number Theory 28A78, 28A80 (primary), 11C08, 42B05, 11B75 (secondary) |
| url | https://arxiv.org/abs/2509.02882 |