The Endpoint Cardinality of Discrete Cube Skeleta
Abstract
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 ≤ k < n, the answer for
N centers is N^{1−(n−k)/(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 ≥ 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^{7/8} points.
contains a filled axis-parallel cube skeleton about every point of some
N-point set of centers. For fixed integers 0 ≤ k < n, the answer for
N centers is N^{1−(n−k)/(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 ≥ 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^{7/8} points.
