标题： Khintchine二分法和Schmidt估计对于$\mathbb{R}^d$上的自相似测度 显示英文标题

标题： Khintchine dichotomy and Schmidt estimates for self-similar measures on $\mathbb{R}^d$

摘要： 我们将在度量丢番图逼近中扩展Khintchine和Schmidt的经典定理到自相似测度在$\mathbb{R}^d$上的上下文中。为此，我们建立了相关随机游走在$\text{SL}_{d+1}(\mathbb{R})/\text{SL}_{d+1}(\mathbb{Z})$上的有效等分布。这推广了我们之前的工作，该工作要求$d=1$并限制了Schmidt类型的计数估计仅适用于衰减足够快的近似函数。新颖的技术包括尽管存在代数障碍仍对相关随机游走采用的自举方案，以及对Dani对应关系的精细处理。我们还建立了自相似测度在$\mathbb{R}^d$的代数子簇附近的非集中性质。 显示英文摘要

摘要： We extend the classical theorems of Khintchine and Schmidt in metric Diophantine approximation to the context of self-similar measures on $\mathbb{R}^d$. For this, we establish effective equidistribution of associated random walks on $\text{SL}_{d+1}(\mathbb{R})/\text{SL}_{d+1}(\mathbb{Z})$. This generalizes our previous work which requires $d=1$ and restricts Schmidt-type counting estimates to approximation functions which decay fast enough. Novel techniques include a bootstrap scheme for the associated random walks despite algebraic obstructions, and a refined treatment of Dani's correspondence. We also establish non-concentration properties of self-similar measures near algebraic subvarieties of $\mathbb{R}^d$.