math.NT — Number Theory
Steinhaus Piercing with Slack: The Exact Asymptotic Constant
Let s(d) be the largest order N for which the first n + d points of a sequence in [0,1) meet every uniform n-cell for all n ≤ N. We prove s(d) = d/(1 − log 2) + O(√d), determining the limit asked for by Anholcer et al. and disproving their conjectured coefficient. For strong piercing, which requires meeting every interval of length 1/n, we obtain an exact finite harmonic capacity and rational constructions. A Dirichlet covering lemma and a packing estimate for disjoint gaps transfer the capacity bound to ordinary grids. We also solve the finite largest-gap minimax problem, establish rigidity for infinite heap refinements, and extend the asymptotic estimate to affine arrival budgets. Ordinary near-extremizers satisfy universal laws for root sizes and first-split times, with a corresponding entropy correction. An explicit finite obstruction shows why root constraints alone do not characterize ordinary piercing.