Polynomial-order oscillations in geometric discrepancy

TL;DR

This paper explores the behavior of the homothetic quadratic discrepancy for planar convex bodies, revealing that its optimal growth rate can oscillate between logarithmic and polynomial orders depending on the shape's geometry.

Contribution

The authors demonstrate that the optimal homothetic quadratic discrepancy can exhibit prescribed polynomial oscillations, extending previous results on its growth rates for convex polygons and smooth bodies.

Findings

Optimal discrepancy oscillates between log N N^{1/2} for certain shapes.

Constructed convex bodies with boundary designs causing oscillations in discrepancy growth.

Established polynomial-order oscillations in discrepancy growth for  range N^lphaor  (2/5, 1/2).

Abstract

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*} \mathcal{D}(\mathcal{P},\, C)=\sum_{\mathbf{p}\in\mathcal{P}}\sum_{\mathbf{n}\in\mathbb{Z}^2}\mathbf{1}_C(\mathbf{p}+\mathbf{n})-N|C|, \end{equation*} and one may estimate how $\mathcal{P}$ deviates from uniformity by averaging the latter quantity over a family of sets. When considering quadratic averages over translated and dilated copies of $C$, one gets the \textit{homothetic quadratic discrepancy} \begin{equation*} \mathcal{D}_2(\mathcal{P},\, C)=\int_{0}^{1}\int_{[0,1)^2}\left|\mathcal{D}( \mathcal{P},\,\boldsymbol{\tau}+\delta C)\right|^2\,{\rm d}\boldsymbol{\tau}\,{\rm d} \delta. \end{equation*} We investigate the behaviour of the optimal \textit{homothetic quadratic discrepancy}, that is \begin{equation*} \inf_{\# \mathcal{P}=N} \mathcal{D}_2(\mathcal{P},\, C)\quad\text{as}\quad N\to+\infty. \end{equation*} Beck~\cite{MR915529} and Beck and Chen~\cite{MR1489133} showed that the optimal \textit{h.q.d.} of convex polygons has an order of growth of $\log N$, and more recently, Brandolini and Travaglini~\cite{MR4358540} proved that the optimal \textit{h.q.d.} of planar convex bodies with a $\mathcal{C}^2$ boundary has an order of growth of $N^{1/2}$. We show that, in general, a single order of growth for the optimal \textit{h.q.d.} need not exist. First, by an implicit geometric construction of $C$, we obtain prescribed oscillations between $\log N$ and $N^{1/2}$. Second, by a subtler design of $\partial C$ and via Fourier-analytic methods, we obtain prescribed polynomial-order oscillations in the range $N^\alpha$ with $\alpha\in(2/5,1/2)$.