Birkhoff generic points on curves in horospheres

TL;DR

This paper proves that Birkhoff genericity extends from the horospherical subgroup to any non-degenerate analytic curve within it, with applications in Diophantine approximation and number theory.

Contribution

It generalizes Birkhoff genericity from the horospherical subgroup to arbitrary non-degenerate analytic curves in the setting of homogeneous dynamics.

Findings

Birkhoff genericity holds along non-degenerate analytic curves in horospheres.

Density estimates for Dirichlet improvability along typical curves.

Applications to algebraic number approximation and best approximation problems.

Abstract

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).