On the best constant in the finitary Vitali covering lemma for high dimensional cubes
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908581276680192 |
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| author | Dall'Ara, Gian Maria |
| author_facet | Dall'Ara, Gian Maria |
| contents | Let $Γ_d$ be the largest constant such that every finite collection of cubes in $\mathbb{R}^d$ whose sides are parallel to the coordinate axes admits a disjoint sub-collection occupying a fraction $Γ_d$ of its volume. Vitali's greedy algorithm shows that $Γ_d\geq 3^{-d}$, and cutting a cube into its $2^d$ dyadic sub-cubes gives $Γ_d\leq 2^{-d}$. The question of determining the value of $Γ_d$ was first raised by T.~Radó in a 1927 letter to Sierpinski.
In this paper we investigate the asymptotic behavior of $Γ_d$ in the high-dimensional limit. We prove that there exists an absolute constant $c>0$ such that \[
Γ_d\geq c\frac{2^{-d}}{d\log d}
\] in all dimensions $d$, a significant asymptotic improvement of earlier results by R.~Rado (1949) and Bereg--Dumitrescu--Jiang (2010). This gives an answer to problem D6 in Croft--Falconer--Guy's book "Unsolved problems in geometry". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the best constant in the finitary Vitali covering lemma for high dimensional cubes Dall'Ara, Gian Maria Classical Analysis and ODEs Metric Geometry Let $Γ_d$ be the largest constant such that every finite collection of cubes in $\mathbb{R}^d$ whose sides are parallel to the coordinate axes admits a disjoint sub-collection occupying a fraction $Γ_d$ of its volume. Vitali's greedy algorithm shows that $Γ_d\geq 3^{-d}$, and cutting a cube into its $2^d$ dyadic sub-cubes gives $Γ_d\leq 2^{-d}$. The question of determining the value of $Γ_d$ was first raised by T.~Radó in a 1927 letter to Sierpinski. In this paper we investigate the asymptotic behavior of $Γ_d$ in the high-dimensional limit. We prove that there exists an absolute constant $c>0$ such that \[ Γ_d\geq c\frac{2^{-d}}{d\log d} \] in all dimensions $d$, a significant asymptotic improvement of earlier results by R.~Rado (1949) and Bereg--Dumitrescu--Jiang (2010). This gives an answer to problem D6 in Croft--Falconer--Guy's book "Unsolved problems in geometry". |
| title | On the best constant in the finitary Vitali covering lemma for high dimensional cubes |
| topic | Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2510.06817 |