关于矩阵的Folklore集与Dirichlet谱
On the Folklore set and Dirichlet spectrum for matrices
AI总结：
本文研究$\mathbb R^{m\times n}$中既非奇异也非坏逼近的Dirichlet可改进矩阵的Folklore集，证明除特定整数对外其非空，构造具预设Dirichlet常数的子集以提供Dirichlet谱信息，并得豪斯多夫维数下界，补充了先前研究。
摘要：
我们研究了$\mathbb R^{m\times n}$中既非奇异也非坏逼近的Dirichlet可改进矩阵的Folklore集。我们以构造性方式证明，除$\{m,n\}=\{1,1\}$和$\{m,n\}=\{2,3\}$外，所有正整数对$m,n$对应的Folklore集均非空。对于广泛的整数对$(m,n)$，我们构造了Folklore集的子集，其Dirichlet常数具有精确的预设值（在0的某个右邻域内）。这使我们能够提供关于矩阵Dirichlet谱的信息。我们构造的关键技术是首先构建给定丢番图类型的向量，然后证明大多数向矩阵的‘提升’将保留该丢番图类型。这是Moshchevitin为一致逼近引入的方法的变体。我们的技术通常也适用于任意范数。作为推论，我们得到了这些集的豪斯多夫维数的下界。这些结论补充了中间作者（Selecta Math. 2023）、Beresnevich等人（Adv. Math. 2023）和Das等人（Adv. Math. 2024）的先前结果。
英文摘要：
We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and $\{m,n\}=\{ 2,3\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024).
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