Irregularities of distribution and geometry of planar convex sets

TL;DR

This paper extends Roth's theorem to planar convex sets, establishing sharp bounds on distribution irregularities depending on boundary smoothness, with implications for geometric and Fourier analysis.

Contribution

It provides new sharp bounds for distribution irregularities of point sets in relation to convex set boundaries, considering different smoothness conditions.

Findings

For smooth boundaries, the bound is proportional to N^{1/2}.

For less smooth, non-polygonal boundaries, the bound is proportional to N^{2/5}.

Intermediate results interpolate between these bounds.

Abstract

We consider a planar convex body $C$ and we prove several analogs of Roth's theorem on irregularities of distribution. When $\partial C$ is $\mathcal{C}% ^{2}$ regardless of curvature, we prove that for every set $\mathcal{P}_{N}$ of $N$ points in $\mathbb{T}^{2}$ we have the sharp bound \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\geqslant cN^{1/2}\;. \] When $\partial C$ is only piecewise $\mathcal{C}^{2}$ and is not a polygon we prove the sharp bound% \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( \lambda C+t\right) \right) -\lambda ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtd\lambda\geqslant cN^{2/5}. \] We also give a whole range of intermediate sharp results between $N^{2/5}$ and $N^{1/2}$. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of $C$.