A Lower Bound for the Discrepancy of a Random Point Set

We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s \le N$, the point set consisting of $N$ points chosen uniformly at random in the $s$-dimensional unit cube $[0,1]^s$ with probability at least $1-\exp(-\Theta(s))$ admits an axis parallel rectangle $[0,x] \subseteq [0,1]^s$ containing $K \sqrt{sN}$ points more than expected. Consequently, the expected star discrepancy of a random point set is of order $\sqrt{s/N}$.Comment: 7 page

On the Discrepancy of Jittered Sampling

We study the discrepancy of jittered sampling sets: such a set $\mathcal{P} \subset [0,1]^d$ is generated for fixed $m \in \mathbb{N}$ by partitioning $[0,1]^d$ into $m^d$ axis aligned cubes of equal measure and placing a random point inside each of the $N = m^d$ cubes. We prove that, for $N$ sufficiently large, $$\frac{1}{10}\frac{d}{N^{\frac{1}{2} + \frac{1}{2d}}} \leq \mathbb{E} D_N^*(\mathcal{P}) \leq \frac{\sqrt{d} (\log{N})^{\frac{1}{2}}}{N^{\frac{1}{2} + \frac{1}{2d}}},$$ where the upper bound with an unspecified constant $C_d$ was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in $N$. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime $N \gtrsim d^d$. We also prove a partition principle showing that every partition of $[0,1]^d$ combined with a jittered sampling construction gives rise to a set whose expected squared $L^2-$discrepancy is smaller than that of purely random points

Minimal discrepancy points on the sphere

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2019, Director: Jordi Marzo Sánchez[en] Let $\mathbb{S}^{k}=\left\{x \in \mathbb{R}^{k+1} ;\|x\|=1\right\}$ be the unit sphere in $\mathbb{R}^{k+1}$ and consider the normalized surface area measure $\sigma^{*}$. It is well known that a set of $n$ points $x_{1}, \ldots, x_{n} \in \mathbb{S}^{k}$ is asymptotically uniformly distributed, i.e., the probability measure $\frac{1}{n} \sum_{j=1}^{n} \delta_{x_{j}}$ converges in the weak- $^{*}$ topology to $\sigma^{*},$ if and only if the spherical cap discrepancy of the set $P=\left\{x_{1}, \ldots, x_{n}\right\},$ defined as $$\mathbb{D}_{n}(P)=\sup _{C(x, t) \subset S^{k}}\left|\operatorname{card}(P \cap C(x, t))-n \sigma^{*}(C(x, t))\right|$$ where $$C(x, t)=\left\{y \in \mathbb{S}^{k} ;\langle x, y\rangle \leq t\right\}$$ is a spherical cap on $\mathbb{S}^{k}$ with $x \in \mathbb{S}^{k}$ and $-1 \leq t \leq 1,$ converges to zero when $n \rightarrow \infty$ It is therefore natural to consider the velocity of this convergence as a measure of the distribution of the sets $P$ In a couple of papers from $1984,$ J. Beck established the following results, which give the best bounds known up to now, [5,6]: - There exist $n$ -element sets of points $P \subset \mathbb{S}^{k}$ such that $$\mathbb{D}_{n}(P) \lesssim n^{\frac{1}{2}-\frac{1}{2 k}} \sqrt{\log n}$$ - For any $n$ -element set of points $P \subset \mathbb{S}^{k}$ $$\mathbb{D}_{n}(P) \gtrsim n^{\frac{1}{2}-\frac{1}{2 k}}$$ It is not known if any of these bounds is sharp. The lower bound uses Fourier analysis and the upper bound some random configurations in regular area partitions of the sphere. The main objective of this master thesis is to study J. Beck's work and the "almost tight" examples obtained through determinantal point processes [9]

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are $n$ elements and $m$ sets and each element lies in $t$ randomly chosen sets. In this setting, Ezra and Lovett showed an $O((t \log t)^{1/2})$ discrepancy bound in the regime when $n \leq m$ and an $O(1)$ bound when $n \gg m^t$. In this paper, we give a tight $O(\sqrt{t})$ bound for the entire range of $n$ and $m$, under a mild assumption that $t = \Omega (\log \log m)^2$. The result is based on two steps. First, applying the partial coloring method to the case when $n = m \log^{O(1)} m$ and using the properties of the random set system we show that the overall discrepancy incurred is at most $O(\sqrt{t})$. Second, we reduce the general case to that of $n \leq m \log^{O(1)}m$ using LP duality and a careful counting argument

Discrepancy convergence for the drunkard's walk on the sphere

We analyze the drunkard's walk on the unit sphere with step size theta and show that the walk converges in order constant/sin^2(theta) steps in the discrepancy metric. This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at http://www.math.hmc.edu/~su/papers.htm