多项式增长的集合与测度的逆定理
Inverse theorems for sets and measures of polynomial growth
摘要：
本文给出了任意群$G$的有限子集$A$的结构描述，对于有界的$d$和充分大的$n$，这些子集满足多项式增长条件$|A^n| \leq n^d |A|$，证明了这些子集在一定的精确意义下受陪集零级数（有界平移数）的控制.这种描述恢复了以前的一些结果Breuillard绿色陶和Breuillard Tointon关于多项式增长的集合，我们也能够描述随后的增长$|A^m|$相当明确的$m \geq n$，至少当$A$是一个对称的邻居的身份。我们还得到了对称概率测度$μ$的一个类似的描述，它的$n$ -倍卷积$μ^{*n}$服从条件$\| μ^{*n} \|_{\ell^2}^{-2} \leq n^d \|μ\|_{\ell^2}^{-2}$ .在阿贝尔的情况下，这种描述恢复逆Littlewood-Offord定理的Nguyen-Vu，并给出了一个变种的最近nonabelian逆Littlewood-Offord定理的Tiep-Vu。我们建立这些结果的主要工具是Breuillard，Green和作者描述近似群结构的逆定理。
英文摘要：
We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard-Green-Tao and Breuillard-Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of $|A^m|$ fairly explicitly for $m \geq n$, at least when $A$ is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures $μ$ whose $n$-fold convolutions $μ^{*n}$ obey the condition $\| μ^{*n} \|_{\ell^2}^{-2} \leq n^d \|μ\|_{\ell^2}^{-2}$. In the abelian case, this description recovers the inverse Littlewood-Offord theorem of Nguyen-Vu, and gives a variant of a recent nonabelian inverse Littlewood-Offord theorem of Tiep-Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.
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。
