Let \(\{a_t: t \in \mathbb {R}\}< {{\,\textrm{SL}\,}}_d(\mathbb {R})\) be a diagonalizable subgroup whose expanding horospherical subgroup \(U < {{\,\textrm{SL}\,}}_d(\mathbb {R})\) is abelian. By the Birkhoff ergodic theorem, for any \(x \in {{\,\textrm{SL}\,}}_d(\mathbb {R})/{{\,\textrm{SL}\,}}_d(\mathbb {Z})\) and for almost every point \(u \in U\) the point ux is Birkhoff generic for \(a_t\) when \(t \rightarrow \infty \) . 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).
