We show that if A is a finite K-approximate subgroup of an s-step nilpotent group then there is a finite normal subgroup $H \subset {A^{{K^{{O_s}(1)}}}}$ modulo which ${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$ contains a nilprogression of rank at most ${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$ and size at least $\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.

中文翻译：

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幂零弗莱曼定理中的多对数界

我们证明如果一种是一个有限的ķ- 的近似子群s-step 幂零群，则有一个有限正规子群$H \subset {A^{{K^{{O_s}(1)}}}}$取模${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$最多包含 nilprogression 等级${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$和大小至少$\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$. 这部分概括了 Sanders 在阿贝尔情况下获得的接近最优边界，并改进了边界并简化了作者早期结果的阐述。结合 Breuillard-Green、Breuillard-Green-Tao、Gill-Helfgott-Pyber-Szabó 和作者的结果，这导致在剩余幂零群和某些有界度线性群中的 Freiman 型定理的秩界得到改善。