Effective equidistribution on adelic quotients

Finite volume orbits on homogeneous spaces are often intimately connected to number theoretical problems, which allows (unipotent) dynamics to be used for these problems. The theorems of Ratner and the corollary by Mozes-Shah have two disadvantages. First they are usually ineffective and second they require a splitting condition at a fixed place when applied to an adelic homogeneous space. We present a theorem that is  completely quantitative and also dispenses with the splitting condition. This is a recent joint work with G. Margulis, A. Mohammadi, and A. Venkatesh.

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