Polynomial effective density in quotients of $\mathbb H^3$ and $\mathbb H^2\times\mathbb H^2$

Abstract

We prove effective density theorems, with a polynomial error rate, for orbits of the upper triangular subgroup of SL2(R) in arithmetic quotients of SL2(C) and SL2(R) × SL2(R).The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space. E.L. acknowledges support by ERC 2020 grant HomDyn (grant no. 833423). A.M. acknowledges support by the NSF grant DMS-1764246.

Polynomial effective equidistribution

“…The proof of this proposition is significantly more delicate than that of the "toy version" of a shifted curve, and relies on an adaptation of a projection theorem due to Käenmäki, Orponen, and Venieri [KOV17], based on the works of Wolff [Wol00], Schlag [Sch03], and [Zah12], in conjunction with a sparse equidistribution argument due to Venkatesh [Ven10]. These elements also played a crucial role in previous work by E.L. and A.M. [LM21] regarding quantitative density for the action of AU on the spaces we consider here.…”

“…The quantitative decay of correlation can be used to establish quantitative results regarding the equidistribution of translates of pieces of an N -orbit. Specifically we employ the results in [KM96], but there is rich literature around the subject; a more complete list of such papers can be found in [LM21,§1]. Now let ξ : [0, 1] → r be a smooth curve.…”

“…In this paper we announce a quantitative equidistribution result for orbits of a one parameter unipotent group on quotients G/Γ where G is either SL 2 (C) or SL 2 (R) × SL 2 (R) with a polynomial error rate, which is the first quantitative equidistribution statement for individual orbits of unipotent flows on quotients of semi-simple groups beyond the horospherical case. Our approach builds on the paper [LM21] by the first two authors, where an effective density result with a polynomial rate for orbits of a Borel subgroup of a subgroup H ≃ SL 2 (R) of G was proved.…”

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

“…for given T > 0. Moreover, effective versions of equidistribution of U-orbits have many applications to number theory, see [LM14], [EM22], [LM21], [LMW22], [Ven10], [NV21], [EMV09], [CY19], [BV16] and references therein for details. As a result, proving effective versions of Ratner's theorem has attracted much attention and has been a major challenge in homogeneous dynamics.…”

“…Lindenstrauss and Margulis proved [LM14] an effective density result for unipotent orbits in SL(3, R)/SL(3, Z) and applied it to prove an effective version of Oppenheim's conjecture. Recently, Lindenstrauss and Mohammadi [LM21] established an optimal effective density result or unipotent orbits in G/Γ for G = SL(2, R) × SL(2, R) and SL(2, C).…”