解析 pro-$p$ 群的下 $p$-级数与 Hausdorff 维数
The lower $p$-series of analytic pro-$p$ groups and Hausdorff dimension
AI总结：
本文针对 $p$-adic 解析 pro-$p$ 群构造近似级数，得到一致有界偏差，引入有理不变量，证明其 Hausdorff 谱离散且至多含 $2^d$ 个有理数。
摘要：
设 $G$ 是维数为 $d$ 的 $p$-adic 解析 pro-$p$ 群。我们构造了一个近似级数，它按层规则地下降，并且其各项与下 $p$-级数的偏差是一致有界的。这揭示了一组新的有理不变量，它们典范地关联于 $G$，给出了上述一致界，并限制了 $G$ 的闭子群关于下 $p$-级数的 Hausdorff 维数可能取的值。特别地，$G$ 关于下 $p$-级数的 Hausdorff 谱是离散的，且至多包含 $2^d$ 个有理数。
英文摘要：
Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set of rational invariants, canonically associated to $G$, that yield the aforementioned uniform bound and that restrict the possible values for the Hausdorff dimensions of closed subgroups of $G$ with respect to the lower $p$-series. In particular, the Hausdorff spectrum of $G$ with respect to the lower $p$-series is discrete and consists of at most $2^d$ rational numbers.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
