论文标题
关于度量近似亚组
On metric approximate subgroups
论文作者
论文摘要
令$ g $为一个公制$ \ mathrm {d} $在左右翻译下不变的集团，让$ \ bar {\ mathbb {d}} _ r $是身份周围的半径$ r $的球。 a $（k，r）$ - 公制近似子组是对称子集$ x $ $ g $的$ g $，因此，成对产品集$ xx $最多由$ k $ translase of $ x \ bar {\ mathbb {d}}} _ r $。该概念是在Arxiv：Math/0601431中引入的，以及用于离散组的版本（近似子组）。在Arxiv：0909.2190中，由于离散的情况显示，在$ x $有限但大的渐近极限中，“近似值”（或需要多个翻译）可以归因于一个规范相关的谎言组。在这里，我们在公制设置中证明了类似的结果，在$ x $更换有限的$ x $的某个有限涵盖假设下。特别是，如果$ g $具有有限的指数，我们表明任何$（k，r）$ - 公制近似子组接近适当$ r'$的$（1，r'）$ - 公制近似子组。
Let $G$ be a group with a metric $\mathrm{d}$ invariant under left and right translations, and let $\bar{\mathbb{D}}_r$ be the ball of radius $r$ around the identity. A $(k,r)$-metric approximate subgroup is a symmetric subset $X$ of $G$ such that the pairwise product set $XX$ is covered by at most $k$ translates of $X\bar{\mathbb{D}}_r$. This notion was introduced in arXiv:math/0601431 along with the version for discrete groups (approximate subgroups). In arXiv:0909.2190, it was shown for the discrete case that, at the asymptotic limit of $X$ finite but large, the "approximateness" (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on $X$ replacing finiteness. In particular, if $G$ has bounded exponent, we show that any $(k,r)$-metric approximate subgroup is close to a $(1,r')$-metric approximate subgroup for an appropriate $r'$.
