to each integral vector v in $\mathbb{z}^n$, we attach several natural objects of geometric/arithmetic nature. for example:

1. the direction of v (i.e., its radial projection to the unit sphere),
2. the orthogonal lattice to v (i.e., the proper rescaling of the lattice of integral points in the orthogonal hyperplane to v),
3. the residue class of v modulo a fixed integer k.

each of these objects resides in a natural “homogeneous space” which supports a “uniform probability measure”. this allows one to ask statistical questions regarding these objects as v varies in some meaningful set of integral vectors. i will survey some classical and more recent results along these lines where there are limit laws governing the statistics. in some cases one obtains the uniform measure as the limit and in some cases a non-uniform limit. interesting examples include the integral points on quadratic surfaces and the sequence of “best approximations” of an irrational line. in the talk i will try to explain how homogeneous dynamics can be used to tackle such questions.