In the talk I will discuss certain diophantine conditions, and in particular I will show how a new Margulis’ type inequality for translates of horospherical orbits helps verify such conditions, leading to a quantified equidistribution result for a large class of points, akin to the results of A. Strombergsson dealing with SL2 case. In particular we deduce a fully effective quantitative equidistribution statement for horospherical trajectories of lattices defined over number fields, without pertaining to Roth’s/strong subspace theorem.
If time permits, I will discuss relations of this result to the recent work about effective linearization by Lindenstrauss-Margulis-Mohammadi-Shah.