A Characterization Theorem for the L2 -Discrepancy of Integer Points in Dilated Polygons

Let C be a convex d-dimensional body. If ρ is a large positive number, then the dilated body ρC contains ρd| C| + O(ρd-1) integer points, where | C| denotes the volume of C. The above error estimate O(ρd-1) can be improved in several cases. We are interested in the L2-discrepancy DC(ρ) of a copy of ρC thrown at random in Rd. More precisely, we consider [Equation not available: see fulltext.]where Td= Rd/ Zd is the d-dimensional flat torus and SO(d) is the special orthogonal group of real orthogonal matrices of determinant 1. An argument of Kendall shows that DC(ρ)≤cρ(d-1)/2. If C also satisfies the reverse inequality DC(ρ)≥c1ρ(d-1)/2, we say that C is L2-regular. Parnovski and Sobolev proved that, if d> 1 , a d-dimensional unit ball is L2-regular if and only if d≢1(mod4). In this paper we characterize the L2-regular convex polygons. More precisely, we prove that a convex polygon is not L2-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.