Polynomial-order oscillations in geometric discrepancy

TL;DR

This work shows that the optimal growth rate of the planar homothetic quadratic discrepancy $\mathcal D_2$ for convex bodies can exhibit prescribed oscillations between logarithmic and polynomial scales, challenging the notion of a single universal rate. It introduces two independent constructions: a geometric, implicit nested-convex-body method that realizes jumps among $\log N$, $N^{2/5}$, and $N^{1/2}$, and a Fourier-analytic method that directly builds boundary features to realize polynomial oscillations with exponents $\alpha\in(2/5,1/2)$. The results demonstrate that, for general convex bodies, the optimal h.q.d. need not fixate on a single order of growth and can be steered across a continuum of polynomial orders. This has implications for understanding irregularities of distribution in geometric contexts and highlights the interplay between boundary geometry, Fourier mass distribution, and discrepancy statistics.

Abstract

Let $C\subset\mathbb{R}^2$ be a convex body, and let $\mathcal{P}\subset[0,1)^2$ be a set of $N$ points. The discrepancy of $\mathcal{P}$ with respect to $C$ is defined as \begin{equation*} \mathcal{D}(\mathcal{P},\, C)=\sum_{\mathbf{p}\in\mathcal{P}}\sum_{\mathbf{n}\in\mathbb{Z}^2}\mathds{1}_C(\mathbf{p}+\mathbf{n})-N|C|. \end{equation*} A standard approach for estimating how a point distribution deviates from uniformity is to average the latter quantity over a family of sets; in particular, when considering quadratic averages over translations and dilations, one obtains \begin{equation*} \mathcal{D}_2(\mathcal{P},\, C)=\int_{0}^{1}\int_{[0,1)^2}\left|\mathcal{D}( \mathcal{P},\,\boldsymbolτ+δC)\right|^2\,{\rm d}\boldsymbolτ\,{\rm d} δ. \end{equation*} This paper concerns the behaviour of optimal \textit{homothetic quadratic discrepancy} \begin{equation*} \inf_{\# \mathcal{P}=N} \mathcal{D}_2(\mathcal{P},\, C)\quad\text{as}\quad N\to+\infty. \end{equation*} Beck and Chen \cite{MR1489133} showed that the optimal \textit{h.q.d.} of convex polygons has an order of growth of $\log N$. More recently, Brandolini and Travaglini \cite{MR4358540} showed that the optimal \textit{h.q.d.} of planar convex bodies has an order of growth of $N^{1/2}$ if their boundary is $\mathcal{C}^2$, and of $N^{2/5}$ if their boundary is only piecewise-$\mathcal{C}^2$ and not polygonal. We show that, in general, no order of growth is required. First, by an implicit geometric construction, we show that one can obtain prescribed oscillations from a logarithmic order of $\log N$ to polynomial orders of $N^{2/5}$ and $N^{1/2}$, and vice versa. Secondly, by using Fourier-analytic methods, we show that prescribed polynomial-order oscillations in the range $N^α$ with $α\in(2/5,1/2)$ are achievable.

Figures (4)

• Figure 1: The construction in the proof of Theorem \ref{['main1']}.
• Figure 2: The chord in Definition \ref{['Corde1']}.
• Figure 3: A close-up look at $P$, in which different auxiliary monomial curve segments are displayed in different colours.
• Figure 4: The auxiliary tools we constructed.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.3: Brandolini-Travaglini
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Definition 3.1
• Lemma 3.2
• Proposition 3.3
• Lemma 3.4