Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation

TL;DR

This paper proves that for almost all points on certain analytic curves in R^k, Dirichlet's approximation theorem cannot be improved, using ergodic theory and equidistribution of flows on homogeneous spaces.

Contribution

It establishes the equidistribution of expanding translates of curves on homogeneous spaces, leading to a new proof of optimal Diophantine approximation results.

Findings

Almost all points on non-affine analytic curves satisfy optimal Dirichlet approximation.

Equidistribution of curve trajectories under partially hyperbolic flows is proven.

Ergodic properties of unipotent flows are utilized to derive number-theoretic results.

Abstract

We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces.