{"id":"hexagon/2610.00243","pdf":"https://hexagonmath.org/pdf/2610.00243v1","url":"https://hexagonmath.org/abs/2610.00243v1","tags":["math.CO","msc:11K38","msc:05D99","Steinhaus problem","entropy","harmonic capacity","interval refinement","irregularities of distribution","piercing sequences"],"score":0,"title":"Steinhaus Piercing with Slack: The Exact Asymptotic Constant","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Anton Joha"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable","ChatGPT"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}