Description
Title: Asymptotics of the number of points of symplectic lattices in subsets of Euclidean spaces
Abstract: It is well known that a "good" large subset of the Euclidean space contains approximately as many lattice points as its volume. This need not hold for a general subset. On the other hand, a classical theorem of Siegel asserts that for any subset of positive measure, the "average" number of points (in an appropriate sense) of a general unimodular lattice contained in it, equals the measure of the set. In place of the average over the entire space of lattices one may also ask for analogous results for smaller subclasses. In a recent work with Jayadev Athreya, we explored this issue, with some modifications that place the problem in perspective, for the case of symplectic lattices, viz. lattices (in even-dimensional spaces) obtained from the standard lattice under symplectic transformations. In this talk I shall describe the overall asymptotics in this case, together with the historical background of the results and techniques involved.