S-arithmetic spaces are modules over a ring given by the product of a real field and finite number of distinct p-adic fields. One can define lattices on S-arithmetic spaces, both as free modules and as discrete subgroups with finite covolume, like lattices in real vector spaces. Hence one can extend problems related to lattices (such as counting lattice points in the specific domains) in number theory and homogeneous dynamics to S-arithmetic spaces.
In this talk, we will consider the notion of primitive lattice points, define the primitive Siegel transform, which can be convenient especially for the 2-“dimensional” case (compare to the Siegel transform over the whole lattice points), and formulate its first and second moment formulas. Among applications, we will see the logarithm law for unipotent flows of the special linear group over an S-arithmetic ring.