We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $|\chi(g)| \le \chi(1)^{\alpha_g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.

中文翻译：

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Lie 类型有限群的字符边界

我们为 Lie 类型的有限群 $G$ 建立了字符值和字符比率的新边界，这比以前已知的边界强得多，并且在许多情况下是最好的。这些边界的形式为 $|\chi(g)| \le \chi(1)^{\alpha_g}$，并产生了多种应用，例如覆盖这些组上随机游走的数字和混合时间。特别是我们推导出，如果$G$是$n$维上的经典群，那么在$G$和$g\inG$的某些条件下，$G$上的随机游走与作为生成集的 $g$ 的共轭类是（最多一个小的乘法常数）$n/s$，其中 $s$ 是 $g$ 的支持。