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We present a new argument in the study of positive entropy measures forhigher rank diagonalisable actions. The argument relies on a quantitative formof recurrence along unipotent directions (that are not known to preserve themeasure). Using this argument we prove a classification of positive entropymeasures for any higher rank action on an irreducible arithmetic quotient of aform of $SL_2$. We also provide an Adelic version of this classification resultwhere no entropy assumption is needed. These results can also be used to provenew results regarding Diophantine approximations of integer multiples of anarbitrary element $\alpha\in\mathbb{R}$.

Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations