We will discuss applications of a discrepancy trick, which sometimes can be used to prove quantitative disjointness results. As an illustration, consider the following example. Fix an imaginary quadratic number field and consider a "diagonal" measure on a product of the unit tangent bundles of two modular curves given by the push-forward under two multivariate monomials of the uniform measure on two copies of the class group. If the monomials are independent, assuming GRH for the Dedekind zeta function, this diagonal measure is quantitatively equidistributed in the product in terms of the size of the discriminant of the quadratic field.

This version of the result implies (conditional) bounds on the size of torsion subgroups of the class group, which was proven previously by Ellenberg and Venkatesh. The main dynamical theorem has several applications and, for example, can be used to prove for most dimensions quantitatively the equidistribution of the triples consisting of a rational subspace, the shape of the integer lattice in this subspace, and the shape of the integer lattice in the orthogonal complement.

This is joint work with Menny Aka, Manfred Einsiedler, Philippe Michel, and Andreas Wieser.