Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

TL;DR

This paper proves that a multiplicative Dirichlet theorem cannot be improved for almost all points on certain analytic curves in R^k, using advanced techniques from homogeneous dynamics and unipotent flows.

Contribution

It establishes the optimality of Dirichlet's theorem on curves in R^k and introduces new methods involving linear dynamics and Ratner's theorem.

Findings

Almost all points on analytic curves satisfy the Dirichlet bound.

Sequences of expanding translates of curves become equidistributed.

New linearization techniques are developed for SL(m,R) dynamics.

Abstract

We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).