Effective equidistribution of expanding translates in the space of affine lattices

Abstract

We prove a polynomially effective equidistribution result for expanding translates in the space of d-dimensional affine lattices for any d ě 2.

Polynomial effective equidistribution

“…When G is unipotent, an effective equidistribution result for unipotent flows on G/Γ was given by Green and Tao in [GT12], and was a key ingredient in a series of works by Green, Tao and Ziegler about linear equations in primes. In the case of quotients of the skew product G = SL 2 (R) ⋉ R 2 , Strombergsson [Str15] has an effective equidistribution result for one parameter unipotent orbits (which are not horospheric in G, but project to a horospheric group on SL 2 (R)), and this has been generalized by several authors, in particular by Wooyeon Kim [Kim21] (using a completely different argument) to SL n (R) ⋉ R n . Moreover there is an important work of Einsiedler, Margulis and Venkatesh [EMV09] showing that periodic orbits of semisimple subgroups H of a semisimple group G are quantitatively equidistributed in an appropriate homogeneous subspace of G/Γ if Γ is a congruence lattice and H has finite centralizer in G. Subsequently Einsiedler, Margulis, Venkatesh and the second named author by using Prasad's volume formula and a more adelic view point were able to prove such an equidistribution result for periodic orbits of maximal semisimple subgroups of G when the subgroup is allowed to vary [EMMV20] with arithemetic applications.…”

“…There have been several extensions of this result, in particular [HdS19] where a proximality assumption was removed. Kim in [Kim21] used the techniques of [BFLM11] to study SL n (R) ⋉ R n . Our work is also heavily influenced by [BFLM11].…”

Effective version of Ratner's equidistribution theorem for $\mathrm{SL}(3,\mathbb{R})$

“…The reader is also refered to [BV16] for a similar effective equidistibution result which has applications to number theory. Strombergsson's result was recently generalized by Kim [Kim21] to SL(n, R) ⋉ R n /SL(n, Z) ⋉ Z n with the unipotent subgroup being horospherical in the semisimple part. Recently, Lindenstrauss, Mohammadi and Wang [LMW22] established effective Ratner's theorem for unipotent orbits in G/Γ where G = SL(2, R) × SL(2, R) and SL(2, C).…”