返回集合中的乘积集与对称遍历平均的正性
Product sets in sets of returns and positivity of symmetric ergodic averages
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中文总结 AI 辅助
该研究推广了Bergelson等人的结果，证明可和群中足够大的集合包含满足乘积集关系的大子集，还针对有限生成幂零群建立了多项式类似结果。
中文摘要 AI 辅助
我们研究可数群$G$中（可测）返回的集合，即由保测作用产生的形如$\{g\in G:\mu(A\cap T_gA)>0\}$的集合。推广Bergelson的结果，我们证明$G\times G$中的返回集合包含形如$B\times B$的子集，其中$B$相对于合适的“大”概念是大的，这些概念即使对于非可和群也仍有意义。因此，若$G$是可和群，则每个足够大的子集$A\subseteq G\times G$都满足对某个大集合$B\subseteq G$有$B\times B\subseteq AA^{-1}$。我们还研究$G$中的返回集合何时包含大集合$B$的乘积集$BB$。与上述笛卡尔积现象相反，该问题在非阿贝尔群中要微妙得多，且与“对称相关函数”密切相关，即形如$g\mapsto \mu(T_g^{-1}A\cap T_gA)$的函数。我们利用这一关联证明，对于广泛类别的可和群——包括有限生成幂零群和某些可解非幂零群，每个足够大的集合$A\subseteq G$都包含一个大子集$B$，满足$BB\subseteq AA^{-1}$。最后，我们针对有限生成幂零群建立这些结果的多项式类似，推广了Bergelson和Ruzsa的早期工作。
英文摘要
We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:μ(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto μ(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.
