Irregularities of distribution and geometry of planar convex sets

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 $\mathcal{C}$ regardless of curvature, we prove that for every set ${\mathcal{P}}_N$ of $N$ points in ${\mathbb{T}}^2$ we have the sharp bound \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( λC+t\right) \right) -λ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtdλ\geqslant cN^{1/2}\;. \] When $\partial C$ is only piecewise ${\mathcal{C}}^2$ and is not a polygon we prove the sharp bound% \[ \int_{0}^{1}\int_{\mathbb{T}^{2}}\left\vert \mathrm{card}\left( \mathcal{P}_{N}\mathcal{\cap}\left( λC+t\right) \right) -λ^{2}N\left\vert C\right\vert \right\vert ^{2}~dtdλ\geqslant cN^{2/5}. \] 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 $C$.