PreprintCC BY 4.0arXiv
A Sharp Distributional Lower Bound for Planar Discrepancy
Abstract
We prove that the unnormalized discrepancy function of every n-point set in the unit square has absolute value at least c√(1+log n) on a set of measure at least η, with absolute constants c,η>0. Consequently, its Lp-quasinorm is bounded below by c η1/p √(1+log n) for every p>0.
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BibTeX
@misc{hinrichs2026sharp,
title = {{A Sharp Distributional Lower Bound for Planar Discrepancy}},
author = {Aicke Hinrichs},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33085},
eprint = {2610.33085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33085},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Sharp Distributional Lower Bound for Planar Discrepancy
AU - Aicke Hinrichs
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.33085
UR - https://arxiv.org/abs/2610.33085
AB - We prove that the unnormalized discrepancy function of every n-point set in the unit square has absolute value at least c√(1+log n) on a set of measure at least η, with absolute constants c,η>0. Consequently, its L_p-quasinorm is bounded below by c η^1/p √(1+log n) for every p>0.
ER -