Irregularities of distribution and geometry of planar convex sets (Q2071637)

Let \(C\) be a planar convex body and \(\mathcal{P}_N\) be \(N\)-point set in \(\mathbb{T}^2\). Consider the following measure of irregularities of distribution for \(\mathcal{P}_N\): \[\Delta(\mathcal{P}_N)=\int_0^1\int_{\mathbb{T}^2} |\mathrm{card} (\mathcal{P}_N\cap (\tau C+t))-\tau^2N|C||^2dtd\tau .\] It is proved that if the boundary of \(C\) is \(\mathcal{C}^2\) then there exists \(c>0\) such that for any \(\mathcal{P}_N\) we have \[\Delta(\mathcal{P}_N)\geq cN^{1/2}\] and the boundary is sharp. Similar estimates (with smaller degree of \(N\)) is also obtained under more weak conditions on the boundary. For example, is the boundary of \(C\) is piecewise \(\mathcal{C}^2\) and \(C\) is not a polygon, we have \[\Delta(\mathcal{P}_N)\geq cN^{2/5}.\]