Consider the number of lattice points inside a given symmetric set, for example, a ball centered at the origin whose volume is $V$, as a random variable on the space of unimodular lattices. In 1960, Schmidt showed that as dimension goes to infinity, the sequence of these random variables with fixed volume $V$ converges to the Poisson distribution. His work was developed by Södergren (2011) and Strömbergsson-Södergren (2019) in diverse settings.
By joint work with Anish Ghosh and Mahbub Alam, we obtained analogs of their results on 1) the space of unimodular affine lattices; 2) the space of unimodular lattices with a congruence condition. These works heavily rely on the higher moment formulas of Siegel transforms. In this talk, we recall the idea of proofs using the method of moments and how to obtain the higher moment formulas for those two cases.