标题： 从球到开集的维数质量传递原理及其在动力学丢番图逼近中的应用 显示英文标题

标题： A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation

摘要： 伯斯涅维奇和维拉尼的质量传递原理是确定在丢番图逼近中自然出现的$\limsup$集合的豪斯多夫维数/测度的强大机制。 然而，在动力学丢番图逼近的背景下，这一原理通常无法有效应用，因为定义动力学$\limsup$集合的球的半径通常依赖于点$x$的轨道本身。 在本文中，我们发展了一个维数质量传递原理，使我们能够恢复并扩展关于收缩目标问题的经典结果，特别是针对$\beta$变换和高斯映射。 此外，我们的结果表明，相应的$\limsup$集合具有大交集性质。 我们方法的一个潜在有趣的特点是，在许多情况下，收缩目标问题与寻找适当的吉布斯测度密切相关，这可能揭示热力学形式主义与动力学丢番图逼近之间联系的新方面。 显示英文摘要

摘要： The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\limsup$ sets generally depend on the orbit of the point $x$ itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $\beta$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\limsup$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.