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

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