Related papers: A characterization theorem for the $L^{2}$-discrep…

Our main contribution is a concentration inequality for the symmetric volume difference of a $ C^2 $ convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability…

Let A be an n by n matrix with determinant 1. We show that for all n > 2 there exist dimensional strictly positive constants C_n such that the average over the orthogonal group of log rho(A X) d X > C_n log ||A||, where ||A|| denotes the…

Suppose $d+1$ absolutely continuous probability measures $m_0, \ldots, m_d$ on $\mathbb{R}^d$ are given. In this paper, we prove that there exists a point of $\mathbb{R}^d$ that belongs to the convex hull of $d+1$ points $v_0, \ldots, v_d$…

Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density rho=1/2 there are solutions, if holes in the body are allowed. For rho different from 1/2 the body is…

The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether every sequence with values plus or minus one has unbounded discrepancy along all homogeneous arithmetic progressions. We establish weighted variants of this problem,…

This article investigates a new parameter for the high-dimensional regression with noise: the distortion. This latter has attracted a lot of attention recently with the appearance of new deterministic constructions of 'almost'-Euclidean…

The combinatorial discrepancy of arithmetic progressions inside $[N] := \{1, \ldots, N\}$ is the smallest integer $D$ for which $[N]$ can be colored with two colors so that any arithmetic progression in $[N]$ contains at most $D$ more…

In this short note, we prove a version of the Johnson-Lindenstrauss flattening Lemma for point sets taking values in discrete subgroups. More precisely, given $d,\lambda_0,N_0\in\mathbb{N}$ and $\epsilon\in \left(0,\frac{1}{2}\right)$…

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well-known that this…

This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with…

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a $d$-dimensional manifold $\mathcal M$ endowed with a distance $\rho$ so that…

We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on…

We show that for every large enough integer $N$, there exists an $N$-point subset of $L_1$ such that for every $D>1$, embedding it into $\ell_1^d$ with distortion $D$ requires dimension $d$ at least $N^{\Omega(1/D^2)}$, and that for every…

We present an explicit construction of infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots)$ in the $d$-dimensional unit-cube whose periodic $L_2$-discrepancy satisfies $$L_{2,N}^{{\rm…

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any…

Let the map $f:[-1,1]\to[-1,1]$ have a.c.i.m. $\rho$ (absolutely continuous $f$-invariant measure with respect to Lebesgue). Let $\delta\rho$ be the change of $\rho$ corresponding to a perturbation $X=\delta f\circ f^{-1}$ of $f$. Formally…

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $\mathbb Z^d$ from the limiting Wulff shape in arbitrary dimensions. As the number $n$ of elements diverges, we prove that the symmetric difference to…

For any $d\geq 1$, we obtain counting and equidistribution results for tori with small volume for a class of $d$-dimensional torus packings, invariant under a self-joining $\Gamma_\rho<\prod_{i=1}^d\mathrm{PSL}_2(\mathbb{C})$ of a Kleinian…

For a convex body B in three-dimensional Euclidean space, which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus…

In 2004 the second author of the present paper proved that a point set in $[0,1]^d$ which has star-discrepancy at most $\varepsilon$ must necessarily consist of at least $c_{abs} d \varepsilon^{-1}$ points. Equivalently, every set of $n$…