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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.17509 |
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| _version_ | 1866914509112737792 |
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| author | Dall'Ara, Gian Maria Dumitrescu, Adrian |
| author_facet | Dall'Ara, Gian Maria Dumitrescu, Adrian |
| contents | What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Radó in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in $\mathbb{R}^d$, where now we are looking for an optimal constant $F(K)$.
Our utmost interest is for cubes and balls in the high-dimensional regime $d\rightarrow \infty$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+ε_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $ε_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17509 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rado's covering problem for cubes and balls: a semi-survey Dall'Ara, Gian Maria Dumitrescu, Adrian Metric Geometry Discrete Mathematics Classical Analysis and ODEs Combinatorics What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Radó in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in $\mathbb{R}^d$, where now we are looking for an optimal constant $F(K)$. Our utmost interest is for cubes and balls in the high-dimensional regime $d\rightarrow \infty$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+ε_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $ε_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem. |
| title | Rado's covering problem for cubes and balls: a semi-survey |
| topic | Metric Geometry Discrete Mathematics Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2604.17509 |