Related papers: Discrepancy of Arithmetic Progressions in Boxes an…

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} =…

The 2-colouring discrepancy of arithmetic progressions is a well-known problem in combinatorial discrepancy theory. In 1964, Roth proved that if each integer from 0 to N is coloured red or blue, there is some arithmetic progression in which…

The partial coloring method is one of the most powerful and widely used method in combinatorial discrepancy problems. However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number…

We prove that the the discrepancy of arithmetic progressions in the $d$-dimensional grid $\{1, \dots, N\}^d$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}}$. This extends the case $d=1$, which is a celebrated…

A classical problem in combinatorics seeks colorings of low discrepancy. More concretely, the goal is to color the elements of a set system so that the number of appearances of any color among the elements in each set is as balanced as…

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set $[N]=\{1,2,\hdots,N\}$ was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph…

We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every…

Celebrated theorems of Roth and of Matou\v{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta(n^{1/4})$. We study the analogous problem in the $\mathbb{Z}_n$ setting.…

We study the so-called nonconventional averages in the context of lattice spin systems, or equivalently random colourings of the integers. For i.i.d. colourings, we prove a large deviation principle for the number of monochromatic…

The hereditary discrepancy of a set system is a certain quantitative measure of the pseudorandom properties of the system. Roughly, hereditary discrepancy measures how well one can $2$-color the elements of the system so that each set…

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs $H$ of a graph $G$ of the neighborhood set system of $H$ is sandwiched…

Tusn\'ady's problem asks to bound the discrepancy of points and axis-parallel boxes in $\mathbb{R}^d$. Algorithmic bounds on Tusn\'ady's problem use a canonical decomposition of Matou\v{s}ek for the system of points and axis-parallel boxes,…

We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with…

We show that the hereditary discrepancy of homogeneous arithmetic progressions is lower bounded by $n^{1/O(\log \log n)}$. This bound is tight up to the constant in the exponent. Our lower bound goes via proving an exponential lower bound…

Let $(X,\S)$ be a set system on an $n$-point set $X$. The \emph{discrepancy} of $\S$ is defined as the minimum of the largest deviation from an even split, over all subsets of $S \in \S$ and two-colorings $\chi$ on $X$. We consider the…

We study one of the key tools in data approximation and optimization: low-discrepancy colorings. Formally, given a finite set system $(X,\mathcal S)$, the \emph{discrepancy} of a two-coloring $\chi:X\to\{-1,1\}$ is defined as $\max_{S \in…

A result of Spencer states that every collection of $n$ sets over a universe of size $n$ has a coloring of the ground set with $\{-1,+1\}$ of discrepancy $O(\sqrt{n})$. A geometric generalization of this result was given by Gluskin (see…

Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes…

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*}…

The Discrepancy of a hypergraph is the minimum attainable value, over two-colorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a…