  Let $\{a_t: t \in \mathbb{R}\}< SL_{d}(\mathbb{R})$ be a diagonalizable
subgroup whose expanding horospherical subgroup $U < SL_{d}(\mathbb{R})$ is
abelian. By the Birkhoff ergodic theorem, for any $x \in
SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ 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).
