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

Abstract

Let $C$ be a convex $d$-dimensional body. If $\rho$ is a large positive number, then the dilated body $\rho C$ contains ${\rho }^d\left|C\right|+\mathcal{O}\left({\rho }^{d-1}\right)$ integer points, where $\left|C\right|$ 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 \[ D_{C}(ρ):=\left\{ \int_{\mathbb{T}^{d}}\int_{SO(d)}\left\vert \textrm{card}\left( \left( ρσ(C)+t\right) \cap\mathbb{Z}^d\right) - ρ^{d}\left\vert C\right\vert \right\vert ^{2}dσdt\right\} ^{1/2}\ , \] 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 $D_C(\rho )\le c{\rho }^{(d-1)/2}$. If $C$ also satisfies the reverse inequality $D_C(\rho )\ge c_1{\rho }^{(d-1)/2}$, 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 $d\not\equiv 1(\mathrm{mod}4)$. 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.