关于改进狄利克雷定理的零-一定律在任意维度中的情况
A zero-one law for improvements to Dirichlet's theorem in arbitrary dimension
AI总结：
本文通过移除技术条件，证明了在特定条件下ψ-狄利克雷矩阵的零-一定律，并引入了新的方法和条件。
摘要：
设ψ是一个在所有大正实数上定义的连续递减函数。我们说一个实m×n矩阵A是ψ-狄利克雷的，如果对于每个足够大的实数t，都能找到p∈Z^m，q∈Z^n\{0}满足‖Aq-p‖^m < ψ(t)和‖q‖^n < t。通过从Kleinbock-Strömbergsson-Yu所证明的部分零-一定律中移除一个技术条件，我们证明了在ψ(t) < 1/t且tψ(t)递增的情况下，ψ-狄利克雷矩阵的勒贝格测度的零-一定律。实际上，我们证明了在更一般的情况下，将tψ(t)的单调性假设替换为更弱的条件。我们的证明遵循Kleinbock-Strömbergsson-Yu在将问题转化为晶格空间中的缩小目标问题的动态方法。关键的新成分是Kleinbock-Strömbergsson-Yu研究的缩小目标的精心选择的子集家族，以及与相关击中事件相关的短程混合估计。我们的方法也适用于类似的加权问题，其中相关的上确界范数被替换成某些加权准范数。
英文摘要：
Let $ψ$ be a continuous decreasing function defined for all sufficiently large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$, one can find $\mathbf{p} \in \mathbb{Z}^m$, $\mathbf{q} \in \mathbb{Z}^n\setminus\{\mathbf{0}\}$ satisfying $\|A\mathbf{q}-\mathbf{p}\|^m< ψ({t})$ and $\|\mathbf{q}\|^n<t$. By removing a technical condition from a partial zero-one law proved by Kleinbock-Strömbergsson-Yu, we prove a zero-one law for the Lebesgue measure of the set of $ψ$-Dirichlet matrices provided that $ψ(t)<1/t$ and $tψ(t)$ is increasing. In fact, we prove the zero-one law in a more general situation with the monotonicity assumption on $tψ(t)$ replaced by a weaker condition. Our proof follows the dynamical approach of Kleinbock-Strömbergsson-Yu by reducing the question to a shrinking target problem in the space of lattices. The key new ingredient is a family of carefully chosen subsets of the shrinking targets studied by Kleinbock-Strömbergsson-Yu, together with a geometric disjointness argument controlling the short-range correlations for the associated hitting events. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.
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