Dirichlet's theorem on diophantine approximation and homogeneous flows

Details Dirichlet's theorem on diophantine approximation and homogeneous flows Dirichlet's theorem on diophantine approximation and homogeneous flows

2006-12-06

arxiv.org/abs/math/0612171v2

J. Mod. Dyn. 4 (2008), 43-62

Mathematics

Number Theory

24 pages, a revised version

Scientific paper

We show that for any $\epsilon<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $\mu$ there is $\epsilon_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $\epsilon<\epsilon_0$, and for $\mu$-almost every $Y$, Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals. Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space $\text{SL}_{m+n}(\mathbb{R})/\text{SL}_{m+n}(\mathbb{Z})$.

Affiliated with

Also associated with

No associations

Rating

Dirichlet's theorem on diophantine approximation and homogeneous flows does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Dirichlet's theorem on diophantine approximation and homogeneous flows, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dirichlet's theorem on diophantine approximation and homogeneous flows will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-16278