Logarithm laws for one parameter unipotent flows

Abstract

We prove logarithm laws and shrinking target properties for unipotent flows on the homogenous space $Γ\bs G$ 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.

This version corrects a mistake appearing in previous versions of this paper. Explicitly, Lemma 2.8 in the previous version is false as stated, and consequently so is the proof of Lemma 2.9 and Theorem 3. In this version we removed the false statement and proved a modified version of Lemma 2.8 which is sufficient for the proof of Theorem 3