A characterization theorem for the $L^{2}$-discrepancy of integer points in dilated polygons

TL;DR

This paper characterizes which convex polygons are $L^{2}$-regular based on inscribability and symmetry, extending the understanding of discrepancy behavior for integer points in dilated polygons.

Contribution

It provides a complete characterization of $L^{2}$-regular convex polygons, identifying inscribed and symmetric polygons as the key class.

Findings

Convex polygons inscribed in a circle and symmetric about the center are not $L^{2}$-regular.

Non-inscribed or non-symmetric polygons are $L^{2}$-regular.

The characterization extends previous results on $L^{2}$-regularity for higher-dimensional bodies.

Abstract

Let $C$ be a convex $d$-dimensional body. If $\rho$ is a large positive number, then the dilated body $\rho C$ contains $\rho^{d}\left\vert C\right\vert +\mathcal{O}\left( \rho^{d-1}\right) $ integer points, where $\left\vert C\right\vert $ denotes the volume of $C$. The above error estimate $\mathcal{O}\left( \rho^{d-1}\right) $ can be improved in several cases. We are interested in the $L^{2}$-discrepancy $D_{C}(\rho)$ of a copy of $\rho C$ thrown at random in $\mathbb{R}^{d}$. More precisely, we consider \[ D_{C}(\rho):=\left\{ \int_{\mathbb{T}^{d}}\int_{SO(d)}\left\vert \textrm{card}\left( \left( \rho\sigma(C)+t\right) \cap\mathbb{Z}^d\right) - \rho^{d}\left\vert C\right\vert \right\vert ^{2}d\sigma dt\right\} ^{1/2}\ , \] where $\mathbb{T}^{d}=$ $\mathbb{R}^{d}/\mathbb{Z}^{d}$ is the $d$-dimensional flat torus and $SO\left( d\right) $ is the special orthogonal group of real orthogonal matrices of determinant $1$. An argument of D. Kendall shows that $D_{C}(\rho)\leq c\ \rho^{(d-1)/2}$. If $C$ also satisfies the reverse inequality $\ D_{C}(\rho)\geq c_{1} \ \rho^{(d-1)/2}$, we say that $C$ is $L^{2}$\emph{-regular}. L. Parnovski and A. Sobolev proved that, if $d>1$, a $d$-dimensional unit ball is $L^{2}% $-regular if and only if $d\not \equiv 1\ (\operatorname{mod}4)$. In this paper we characterize the $L^{2}$-regular convex polygons. More precisely we prove that a convex polygon is not $L^{2}$-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.