Minimal discrepancy points on the sphere

Abstract

[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].