Birkhoff generic points on curves in horospheres

Abstract

Let $\{a_t:t\in \mathbb{R}\}<S{L}_d(\mathbb{R})$ be a diagonalizable subgroup whose expanding horospherical subgroup $U<S{L}_d(\mathbb{R})$ is abelian. By the Birkhoff ergodic theorem, for any $x\in S{L}_d(\mathbb{R})/S{L}_d(\mathbb{Z})$ and for almost every point $u\in U$ the point $ux$ is Birkhoff generic for $a_t$ when $t\to \infty$. We prove that the same is true when $U$ is replaced by any non-degenerate analytic curve in $U$. This Birkhoff genericity result has various applications in Diophantine approximation. For instance, we obtain density estimates for Dirichlet improvability along typical points on a curve in Euclidean space. Other applications address approximations by algebraic numbers and best approximations (in the sense of Lagarias).

38 pages