Dirichlet's theorem on diophantine approximation and homogeneous flows

Pages: 43 - 62,
Issue 1,
January
2008 doi:10.3934/jmd.2008.2.43

Dmitry Kleinbock - Goldsmith 207, Brandeis University, Waltham, MA 02454-9110, United States (email)

Barak Weiss - Ben Gurion University, Be'er Sheva, 84105, Israel (email)

Abstract:
Given an $m \times n$ real matrix $Y$, an unbounded set $\mathcal{T}$
of parameters $t =( t_1, \ldots,
t_{m+n})\in\mathbb{R}_+^{m+n}$ with $\sum_{i = 1}^m t_i =\sum_{j = 1}^{n} t_{m+j} $ and $0<\varepsilon
\leq 1$, we say that Dirichlet's Theorem can be $\varepsilon$-improved
for $Y$ along $\mathcal{T}$ if for every sufficiently large $\v \in
\mathcal{T}$ there are nonzero $\q \in \mathbb Z^n$ and $\p \in \mathbb Z^m$
such that

$|Y_i\q - p_i| < \varepsilon e^{-t_i}\,$ $i = 1,\ldots, m$

$|q_j| < \varepsilon e^{t_{m+j}}\,$ $j = 1,\ldots, n$

(here $Y_1,\ldots,Y_m$
are rows of $Y$).
We show that for any $\varepsilon<1$ and
any $\mathcal{T}$ 'drifting away from walls', see (1.8),
Dirichlet's Theorem cannot
be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every $Y$.
In the case $m = 1$ we also show that for a large
class of measures $\mu$ (introduced in
[14]) there is $\varepsilon_0>0$ such that for
any drifting away from walls
unbounded $\mathcal{T}$, any $\varepsilon<\varepsilon_0$, and for $\mu$-almost every $Y$,
Dirichlet's Theorem cannot
be $\varepsilon$-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 $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.

Keywords: Diophantine approximation, flows on
homogeneous spaces, equidistribution, quantitative nondivergence,
friendly measures.

Mathematics Subject Classification: Primary: 11J83; Secondary: 11J54, 37A17, 37A45.

Received: December 2006;
Revised:
September 2007;
Available Online: October 2007.