Logarithm laws for one parameter unipotent flows

TL;DR

This paper establishes logarithm laws and shrinking target properties for unipotent flows on certain homogeneous spaces, using estimates of theta series norms, advancing understanding of dynamical behavior in these settings.

Contribution

It introduces new estimates for theta series norms to prove logarithm laws and shrinking target properties for unipotent flows on specific homogeneous spaces.

Findings

Proved logarithm laws for unipotent flows.

Established shrinking target properties in the setting.

Developed methods based on theta series norm estimates.

Abstract

We prove logarithm laws and shrinking target properties for unipotent flows on the homogenous space $\Gamma\bs G$ with $G=\SL_2(\bbR)^{r_1}\times\SL_2(\bbC)^{r_2}$ and $\Gamma\subseteq G$ an irreducible non-uniform lattice. Our method relies on certain estimates for the norms of (incomplete) theta series in this setting.