The Endpoint Cardinality of Discrete Cube Skeleta

We determine the minimum order of a finite lattice set that contains a filled axis-parallel cube skeleton about every point of some $N$-point set of centers. For fixed integers $0\le k<n$, the answer for $N$ centers is $N^{1-(n-k)\mathrm{/}(2n^2)}$, up to constants depending on $n$ and $k$. Thornton proved every smaller exponent and gave a construction of this order; the endpoint lower bound was left open when $k\ge 1$. Our proof combines a midpoint estimate, a labelled form of Shearer’s projection inequality, and a strong induction that balances large and small radii without a dyadic pigeonhole loss. In particular, a lattice set containing a square boundary about each of $N$ centers has at least a constant times $N^{7\mathrm{/}8}$ points.