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On the spectrum of a large subgroup of a semisimple group
Dirichlet's theorem on diophantine approximation and homogeneous flows
1.  Goldsmith 207, Brandeis University, Waltham, MA 024549110 
2.  Ben Gurion University, Be'er Sheva, 84105, Israel 
$Y_i\q  p_i < \varepsilon e^{t_i}\,$ $i = 1,\ldots, m$
$q_j < \varepsilon e^{t_{m+j}}\,$ $j = 1,\ldots, n$
(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 $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.
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