On the best constant in the finitary Vitali covering lemma for high dimensional cubes

TL;DR

The paper improves the asymptotic lower bound for the best Vitali-lemma constant $\Gamma_d$ for axis-parallel cubes in high dimensions by establishing an approximate unit-scale reduction in finite-dimensional normed spaces. Using a two-lemma framework (comparable vs unit radii and lacunary radii) plus a pigeonhole argument, the authors relate $\Gamma_d$ to $\gamma_d$ and optimize a multi-scale parameter $L$ to obtain $\Gamma_d \ge c\,\frac{2^{-d}}{d\log d}$ for all $d$, with an even sharper bound $\Gamma_d \ge e^{-1-\frac{\log\log d}{\log d}+O(1/\log d)}\,\frac{\gamma_d}{d\log d}$. Specializing to $X=\ell_\infty^d$ yields the main result and confirms $\lim_{d\to\infty} \frac{\log(1/\Gamma_d)}{d}=\log 2$, aligning with the conjectured high-dimensional behavior. Numerical analysis supports that the new bound outperforms the classical Vitali constant $3^{-d}$ for $d\ge 14$, providing concrete evidence of significant asymptotic improvement. The work addresses problem D6 in Croft–Falconer–Guy and advances understanding of optimal covering in high dimensions.

Abstract

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".

Theorems & Definitions (12)

• Theorem 1.1: Vitali covering lemma for axes-parallel cubes Vitali1908
• Theorem 1.2: Optimal Vitali covering lemma for axes-parallel congruent cubes
• Theorem 1.3: Almost optimal Vitali covering lemma for high-dimensional axes-parallel cubes
• Theorem 2.1: Vitali covering lemma on a general normed space
• Theorem 2.2: Approximate reduction to unit scale
• Lemma 2.3: Comparable versus unit radii
• proof
• Definition 2.4: ($\lambda,\mu$)-lacunary sets of radii
• Lemma 2.5: Lacunary versus unit radii
• proof