Optimal and typical \(L^2\) discrepancy of 2-dimensional lattices (Q6618004)

For any finite point set \(P\subset [0,1)^d\) its \(L^2\)-discrepancy can be defined as \(D_2(P)=\left(\int_{[0,1]^d} (|P\cap [0,x_1)\times\ldots\times [0,x_d)|-|P|x_1\ldots x_d)^2dx_1\ldots dx_d \right)^{1/2}\). The author considers \(d=2\) and either \(P=L(\alpha,N)=\left\{ \left( \{n\alpha\},\frac{n}{N}\right):0\leq n\leq N-1\right\}\) or \(P=S(\alpha,N)=\left\{ \left( \{\pm n\alpha\},\frac{n}{N}\right):0\leq n\leq N-1\right\}\) and presents a very deep study of \(L^2\)-discrepancies in these cases.\N\NThe obtained results include\N\N1. A characterization (in terms of the continued fraction expansion of \(\alpha\)) of \(\alpha\) (rational and irrational) with optimal order \(\sqrt{\log N}\) of \(D_2(L(\alpha,N))\) and \(D_2(S(\alpha,N))\).\N\N2. Asymptotic formulas for \(D_2(L(\alpha,N))\) and \(D_2(S(\alpha,N))\) for the case than \(\alpha\) is a quadratic irrational.\N\N3. An analogue of Khintchin's theorem for the growth orders of \(D_2(L(\alpha,N))\) and \(D_2(S(\alpha,N))\) for almost all \(\alpha\).\N\N4. Limit distribution theorem for \(5\pi^3\frac{D_2^2(S(\alpha,N))}{\log^2 N}\).