Arithmetic Homogeneous Spaces

I will discuss the following theme: starting with an a priori Diophantine result (typical flavour: integer solutions to such-and-such equation are well-spaced) and turning it into an equidistribution-type statement on a homogeneous space. This (in various guises) appears in much work, starting (at least) with Linnik. I will try to explain by analogy with baby examples on the circle.