Related papers: An L^1 estimate for half-space discrepancy

Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny \neq 0$, $d\geq…

The Hardy-Littlewood inequalities for $m$-linear forms on $\ell_{p}$ spaces are stated for $p>m$. In this paper, among other results, we investigate similar results for $1\leq p\leq m.$ Let $\mathbb{K}$ be $% \mathbb{R}$ or $\mathbb{C}$ and…

We prove that the distortion of any embedding into $L_1$ of the transportation cost space or earth mover distance over a $d$-dimensional grid $\{1,\dots m\}^d$ is $\Omega(\log N)$, where $N$ is the number of vertices and the implicit…

Motivated by multi-objective optimization, we study extrema of a set of N points independently distributed inside the d-dimensional hypercube. A point in this set is k-dominated by another point when at least k of its coordinates are…

Let $\Theta^{(n)}$ be a random vector uniformly distributed on the unit sphere $\mathbb S^{n-1}$ in $\mathbb R^n$. Consider the projection of the uniform distribution on the cube $[-1,1]^n$ to the line spanned by $\Theta^{(n)}$. The…

For a fixed positive integer $d$ and a small real $p>0$, sample a $p$-random subset $A \subseteq \mathbb{Z}_{\geq 0}^d$, and let $S:=\langle A \rangle$ be the generalized numerical semigroup generated by $A$. We show that with high…

We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein…

We prove that if $(\mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $\tau>0$ there is a random subset $\mathcal{Z}$ of $\mathcal{M}$ such that for any pair of points $x,y\in…

We show that the inequality $$ \|D^{k-1}(u-\pi u)\|_{\mathrm{L}^{n/(n-1)}(\mathbb{R}^n)}\leq c\|\mathbb{B}(D) u\|_{\mathrm{L}^1(\mathbb{R}^n)} $$ holds for vector fields $u\in\mathrm{C}^\infty_c$ if and only if $\mathbb{B}$ is canceling.…

We prove that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm. We make two conjectures concerning reverse entropy power inequalities in the log-concave setting and discuss…

By a profound result of Heinrich, Novak, Wasilkowski, and Wo{\'z}niakowski the inverse of the star-discrepancy $n^*(s,\ve)$ satisfies the upper bound $n^*(s,\ve) \leq c_{\mathrm{abs}} s \ve^{-2}$. This is equivalent to the fact that for any…

In this paper, we study the probability distribution of the observable $s = (1/N)\sum_{i=N-N'+1}^N x_i$, with $1 \leq N' \leq N$ and $x_1<x_2<\cdots< x_N$ representing the ordered positions of $N$ particles in a $1d$ one-component plasma,…

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap…

This paper investigates the rate-distortion function, under a squared error distortion $D$, for an $n$-dimensional random vector uniformly distributed on an $(n-1)$-sphere of radius $R$. First, an expression for the rate-distortion function…

For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we…

This article is a continuation of our first work \cite{chaudruraynal:frikha}. We here establish some new quantitative estimates for propagation of chaos of non-linear stochastic differential equations in the sense of McKean-Vlasov. We…

Let $X_1,\dots,X_n$ be i.i.d. log-concave random vectors in $\mathbb R^d$ with mean 0 and covariance matrix $\Sigma$. We study the problem of quantifying the normal approximation error for $W=n^{-1/2}\sum_{i=1}^nX_i$ with explicit…

It is shown that \begin{equation*} \mathsf{P}(\|a_1U_1+\dots+a_nU_n\|>u)\le c\,\mathsf{P}(a\|Z_d\|>u) \end{equation*} for all real $u$, where $U_1,\dots,U_n$ are independent random vectors uniformly distributed on the unit sphere in…

According to the Erd\H{o}s discrepancy conjecture, for any infinite $\pm 1$ sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any $\pm 1$ sequence $(x_1,x_2,...)$ and a discrepancy…

We prove that classical jittered sampling of the $d$-dimensional unit cube does not yield the smallest expected $\mathcal{L}_2$-discrepancy among all stratified samples with $N=m^d$ points. Our counterexample can be given explicitly and…