标题： 关于子流形通用性的$\mathbb{R}^d$-作用和一致乘性丢番图逼近 显示英文标题

标题： Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

摘要： 本文中，我们证明了关于$\mathbb{R}^d$-作用的一个新的遍历定理，涉及沿膨胀子流形的平均值，从而推广了球面平均值的理论。 我们的主要结果是对这类平均值的误差项的一个定量估计，该估计对于光滑函数在作用满足某些有效混合假设时成立。 借助这个定理，我们研究了具有实系数的$(m\times n)$-矩阵的乘法型 Dirichlet 改进性。 特别是，我们证明了对于任意$\varepsilon>0$，几乎所有矩阵都可以被函数$x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$均匀逼近。 此类结果引发了一个问题，可以将其视为乘法丢番图逼近中小伍德猜想的一种加强形式。 显示英文摘要

摘要： In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of such averages valid for smooth functions under some effective mixing assumptions on the action. With the aid of this theorem, we investigate multiplicative-type Dirichlet-improvability for $(m\times n)$-matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ for any $\varepsilon>0$. Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.