Metraipolis › archive › dc_58ee09123a67

Resolving Tusnády's Problem — arXiv:2611.06998

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.06998
License (as archived): CC0-1.0
Archived: 2026-11-06T14:52:54.661Z
SHA-256 of the archived text: fec28a8b9d7365f3e26ae458e66d728211494861e48611c45d1e6177cf30433e
Stable URL of this page: https://metraipolis.com/documento/dc_58ee09123a67

How to cite

Resolving Tusnády's Problem — arXiv:2611.06998 — arXiv (en), https://arxiv.org/abs/2611.06998 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_58ee09123a67 (Metraipolis archive dc_58ee09123a67, sha256 fec28a8b9d7365f3e26ae458e66d728211494861e48611c45d1e6177cf30433e)

BibTeX

@misc{dc58ee09123a67,
  title = {Resolving Tusnády's Problem — arXiv:2611.06998},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.06998},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_58ee09123a67; sha256 fec28a8b9d7365f3e26ae458e66d728211494861e48611c45d1e6177cf30433e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.06998. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove a lower bound of $\Omega_d((\log N)^{d-1/2})$ for the worst-case combinatorial discrepancy of $N$-point sets with respect to anchored boxes in every fixed dimension $d\ge3$. Together with Nikolov's upper bound of the same order and Wei's planar lower bound, this settles Tusnády's problem: the discrepancy of axis-parallel boxes is $\Theta_d((\log N)^{d-1/2})$ in every fixed dimension $d\ge2$. The proof combines smoothed product test functions, sampling discretization, and an entropy argument for conditioned Gaussian vectors.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Classical Analysis and ODEs (math.CA) ; Computational Geometry (cs.CG); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.06998 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-06T14:24:27.851Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).