Andreas Wieser，Hebrew University of Jerusalem

Let $\{a(t):t \in \mathbb{R}\}$ be a diagonalizable subgroup of $SL(d,\mathbb{R})$ for which the expanded horosphere $U$ is abelian. By the Birkhoff ergodic theorem, for any point $x \in SL(d,\mathbb{R})/SL(d,\mathbb{Z})$ and almost every $u \in U$ the point $ux$ is Birkhoff generic for the flow $a(t)$. One may ask whether the same is true when the points in $U$ are sampled with respect to a measure singular to the Lebesgue measure. In this talk, we discuss work with Omri Solan proving that almost every point on an analytic curve within U is Birkhoff generic when the curve satisfies a non-degeneracy condition. This Birkhoff genericity result has various applications in Diophantine approximation. In this talk, we shall use Lagarias' notion of best approximations of vectors as an entry point to the topic. No preliminary knowledge of any of the above notions is assumed.