We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer $$n>1$$n>1 such that the ball of radius n in some Cayley graph of G has cardinality bounded by $$n^{c\log \log n}$$ncloglogn, then G is virtually $$(\log n)$$(logn)-step nilpotent.

中文翻译：

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残差幂零群的近似子群

我们证明残差幂零群 G 的 K 近似子群 A 包含在有限乘幂零子群的有界多个陪集中，其幂零因子是有界步长。结合作者的早期结果，这意味着 A 包含在有界秩和步长的陪集 nilprogression 的有界许多翻译中。边界是有效的并且仅取决于 K；特别是，如果 G 是幂零的，它们不依赖于 G 的步骤。 作为一个应用，我们证明存在一些绝对常数 c，使得如果 G 是一个残差幂零群，并且如果存在整数 $$n>1 $$n>1 使得 G 的某个 Cayley 图中半径为 n 的球具有以 $$n^{c\log \log n}$$ncloglogn 为界的基数，那么 G 实际上是 $$(\log n)$ $(logn)-step 幂零。