DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS

By Dmitry Kleinbock, Barak Weiss. 2006

96 citations recorded by the source corpus.

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 $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d43534a300d1a98f3561d3274fd928be1a45e9c1

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

On effective equidistribution of expanding translates of certain orbits in the space of lattices (2007, methodology): “subset of X. This can be extracted from the next lemma, which is immediate from [ KW , Proposition 2.4] applied to the representations of G on Vj(Rk), j =”
Hausdorff dimension of singular vectors (2016, background): “This result has been generalized in several directions (see for instance [18]) but less is known about the Hausdorff dimension of DIε(d).”
Hausdorff dimension of the set of singular pairs (2007, background): “Further results about singular vectors and divergent trajectories can be found in the papers [10], [11], [23] and [24].”