Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.06998

Resolving Tusnády's Problem

Nicola Wengiel
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove a lower bound of $Ω_d((\log N)^{d-1/2})$ 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 $Θ_d((\log N)^{d-1/2})$ in every fixed dimension $d\ge2$. The proof combines smoothed product test functions, sampling discretization, and an entropy argument for conditioned Gaussian vectors.

Explore related subjects

math.CAcs.CGcs.DSmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nicola Wengiel. 2026-11-04. Resolving Tusnády's Problem. https://arxiv.org/abs/2611.06998

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Group-free radial Kunze-Stein phenomenon on affine buildings

We give a geometric proof of the radial Kunze--Stein phenomenon on the good vertices of a thick, locally finite, regular affine building of irreducible type. The proof uses a positive spherical function, its joint eigenvalues for sphere averages, and the growth of vector-distance spheres. Conjugation by this function and complex interpolation yield a weighted $\ell^2$ estimate and the boundedness of radial operators with $\ell^p$ kernels for $1\leq p<2$. We also present proofs by the averaging operators norms, by the Schur test and by Helgason transform. A further interpolation gives weighted mixed-exponent estimates. No transitivity assumption on the automorphism group is required; in particular the argument applies to buildings of type $\widetilde A_2$ without a Bruhat--Tits realization.

math.CA↗
arXiv2026

Borel Summability for Nonlinear Differential Systems on Characteristic Domains

We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$, where $\hbar$ is a small complex perturbation parameter. We give geometric and analytic conditions under which the formal perturbative solution associated with a fixed regular leading-order branch is locally uniformly Borel summable. This leads to an existence and uniqueness theorem of holomorphic solutions with prescribed perturbative expansion. We give an explicit triangular example and a self-contained application to the singularly perturbed Painlevé I equation.

math.CA↗
arXiv2026

Necessary and sufficient conditions for universality limits

We derive necessary and sufficient conditions for universality limits for orthogonal polynomials associated to a fixed measure on the real line and related systems. One of our results is that the Christoffel--Darboux kernel has sine kernel asymptotics at a point $ξ$, with regularly varying scaling, if and only if the orthogonality measure (spectral measure) has a unique tangent measure at $ξ$ and that tangent measure is the Lebesgue measure. All previously known cases of sine kernel asymptotics, with absolutely continuous or singular measures, have regularly varying scaling. Our work is not limited to bulk universality; we show that the Christoffel--Darboux kernel has a regularly varying scaling limit with a nontrivial limit kernel if and only if the orthogonality measure has a unique tangent measure at $ξ$ and that tangent measure is not a point mass. The possible limit kernels correspond to homogeneous de Branges spaces; in particular, this equivalence completely characterizes several prominent universality classes such as hard edge universality, Fisher--Hartwig singularities, and jump discontinuities in the weights. A key ingredient in the proof is a new correspondence between measures and chains of de Branges spaces, not between Weyl functions and Hamiltonians. In order to handle limits with power law weights, this correspondence goes beyond the more common setting of Poisson-finite measures, and allows arbitrary power bounded measures.

math.CA↗