Related papers: Discrepancy for convex bodies with isolated flat p…

In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of…

The purpose of this paper is to study convex bodies $C$ for which there exists no convex body $C^\prime\subsetneq C$ of the same lattice width. Such bodies shall be called ``lattice reduced'', and they occur naturally in the study of the…

Let $C\subset\mathbb{R}^2$ be a convex body, and for a positive integer $N$, let $\mathcal{P}$ be a configuration of $N$ points in $[0,1)^2$. The discrepancy of $\mathcal{P}$ with respect to $C$ is defined by \begin{equation*}…

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice…

The L_2-discrepancy measures the irregularity of the distribution of a finite point set. In this note we prove lower bounds for the L_2 discrepancy of arbitrary N-point sets. Our main focus is on the two-dimensional case. Asymptotic upper…

We study a version of the Busemann-Petty problem for $\log$-concave measures with an additional assumption on the dilates of convex, symmetric bodies. One of our main tools is an analog of the classical large deviation principle applied to…

The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in $\mathbb R^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it…

The variance of the number of lattice points inside the dilated bounded set rD with random position in R^d has asymptotics r^(d-1) if the rotational quadratic average of the modulus of the Fourier transform of the set is O(r^(-d-1)). The…

Revisiting and extending a recent result of M.Huxley, we estimate the $L^{p}\left( \mathbb{T}^{d}\right) $ and Weak-$L^{p}\left( \mathbb{T}^{d}\right) $ norms of the discrepancy between the volume and the number of integer points in…

Let $C$ be a convex $d$-dimensional body. If $\rho$ is a large positive number, then the dilated body $\rho C$ contains $\rho^{d}\left\vert C\right\vert +\mathcal{O}\left( \rho^{d-1}\right) $ integer points, where $\left\vert C\right\vert $…

Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice…

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic…

We show that the fundamental objects of the $L_p$-Brunn-Minkowski theory, namely the $L_p$-affine surface areas for a convex body, are closely related to information theory: they are exponentials of R\'enyi divergences of the cone measures…

We study the error of the number of points of the lattice $\mathbb{Z}^{d}$ that fall into a dilated and translated hypercube centred around $0$ and whose axis are parallel to the axis of coordinates. We show that if $t$, the factor of…

We study an analogue of the large deviation principle for mixed measures associated with a class of $\log$-concave probability measures whose densities depend on the gauge function of a convex body. For convex bodies in $\mathbb{R}^n$, we…

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…

We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of…

In this paper, we propose the Fourier Discrepancy Function, a new discrepancy to compare discrete probability measures. We show that this discrepancy takes into account the geometry of the underlying space. We prove that the Fourier…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…