Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

Abstract

Abstract. We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should have finite centralizer in the ambient group). The proofs make extensive use of spectral gaps, and also of a closing lemma for such actions.

The quantitative behaviour of polynomial orbits on nilmanifolds

“…5. This result has some faint resemblance to the Szemerédi regularity lemma [30], although with the key difference that our bounds here are all polynomial in nature.…”

“…Indeed the very existence of a discrete and cocompact subgroup Γ guarantees that the lower central series is rational by [4, Th. 5 We refer to the t i as the Mal'cev coordinates of g, and we define the Mal'cev coordinate map ψ = ψ X : G → R m to be the map…”

“…Remark. The reader may wish to compare this with [5], another recent result on quantitative variants of Ratner's theorem.…”

“…Lemma 7. 5. We have a decomposition η(g , g) = η 1 (g) + η 2 (g g −1 ) for all (g , g) ∈ G , where η 1 : G → R/Z is a horizontal character on G, and η 2 : G 2 → R/Z is a horizontal character on G 2 which also annihilates [G,…”

Effective estimates on indefinite ternary forms

“…Another logical choice is measuring the arithmetic complexity of H.x using the discriminant as defined in [ELMV] and [EMV,§17.3]). As shown in [EMV,Prop.…”