Shrinking target horospherical equidistribution via translated Farey sequences

TL;DR

This paper proves shrinking target horospherical equidistribution (STHE) for certain flows on SL(d,R), extending previous results by introducing translated Farey sequences and developing new geometric and dynamical tools.

Contribution

The paper introduces translated Farey sequences and generalizes equidistribution results to translated horospheres, extending known theorems to higher dimensions and new types of targets.

Findings

Established STHE for bounded subsets of horospheres in SL(d,R)

Proved equidistribution of translated Farey sequences

Extended known results from dimension 2 to higher dimensions

Abstract

For a certain diagonal flow on $\operatorname{SL}(d, \mathbb{Z}) \backslash \operatorname{SL}(d, \mathbb{R})$ where $d \geq 2$, we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is shrinking target horospherical equidistribution (STHE), and we show STHE for several types of shrinking targets. Our STHE results extend known results for $d=2$ and $\mathcal{L} \backslash \operatorname{PSL}(2, \mathbb{R})$ where $\mathcal{L}$ is any cofinite Fuchsian group with at least one cusp. The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and Marklof's result on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.