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

Abstract

Let $\Gamma_d$ 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 $\Gamma_d$ of its volume. Vitali's greedy algorithm shows that $\Gamma_d\geq 3^{-d}$, and cutting a cube into its $2^d$ dyadic sub-cubes gives $\Gamma_d\leq 2^{-d}$. The question of determining the value of $\Gamma_d$ was first raised by T.~Rad\'o in a 1927 letter to Sierpinski. In this paper we investigate the asymptotic behavior of $\Gamma_d$ in the high-dimensional limit. We prove that there exists an absolute constant $c>0$ such that \[ \Gamma_d\geq c\frac{2^{-d}}{d\log d} \] 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".