Bounds for Approximate Group Classification

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A finite subset $A$ of a group $G$ is a $K$-approximate group if $1\in A$, $A=A^{-1}$, and $A^2\subset XA$ for some finite $X\subset G$ with $|X|\leq K$. Write $A^m$ for products of $m$ members of $A$. In a group, $P(u_1,\ldots,u_r;L_1,\ldots,L_r)$ consists of words in $u_j^{\pm1}$ in which generator $u_j$ or its inverse occurs in total at most $L_j>0$ times. It is a nilprogression of rank $r$ and step $s$ if every iterated commutator of degree $s+1$ in the generators is the identity; commutator degree counts generator occurrences and $[a,b]=a^{-1}b^{-1}ab$. A coset nilprogression in $G$ is $P=\pi^{-1}(Q)$, where $H$ is a finite normal subgroup of a subgroup $G_0\leq G$, $\pi:G_0\to G_0/H$ is the quotient, and $Q$ is a nilprogression in $G_0/H$.

For $C\geq1$, a nilprogression $Q=P(u_1,\ldots,u_r;L_1,\ldots,L_r)$ is in $C$-normal form if, for all $i<j$ and all sign choices, \[ [u_i^{\pm1},u_j^{\pm1}]\in P\left(u_{j+1},\ldots,u_r;\frac{CL_{j+1}}{L_iL_j},\ldots,\frac{CL_r}{L_iL_j}\right), \] the products $u_1^{m_1}\cdots u_r^{m_r}$ are distinct for integers $|m_i|\leq L_i/C$, and \[ C^{-1}\prod_{i=1}^r(2\lfloor L_i\rfloor+1)\leq|Q|\leq C\prod_{i=1}^r(2\lfloor L_i\rfloor+1). \] An empty-generator progression is $\{1\}$ and an empty product is $1$. A coset nilprogression is in $C$-normal form when its quotient nilprogression is.

Statement recorded in the supplied source.

Make the constants in the arbitrary-group approximate-group classification explicit as functions of $K$. In the full global-group form of the Breuillard--Green--Tao theorem, determine effective functions $r(K),s(K),C(K),D(K)$ such that every finite $K$-approximate group $A$ in every group $G$ has a coset nilprogression $P$ with \[ P\subset A^4,\quad |P|\geq\frac{|A|}{D(K)},\quad \operatorname{rank}(P)\leq r(K),\quad\operatorname{step}(P)\leq s(K), \] in $C(K)$-normal form. Determine as well an explicit covering bound $m(K)$ in the associated finite-by-nilpotent formulation: $A$ is covered by at most $m(K)$ left cosets of some $G_0$, with a finite normal $H\triangleleft G_0$ such that $G_0/H$ has bounded generating rank and nilpotency step, and $A^4$ contains $H$ and a generating set for $G_0$. The dependence must be uniform over arbitrary ambient groups, rather than restricted to nilpotent or fixed-dimensional linear groups. [S2]

Short English statement

Find explicit quantitative bounds for every structural and covering constant in the general classification of finite approximate groups.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 178 (GREEN-090). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Emmanuel Breuillard, Ben Green and Terence Tao, The structure of approximate groups, Publications Mathématiques de l’IHÉS 116 (2012), 115--221; arXiv:1110.5008v2 (2011), Theorem 1.6, Remarks 1.8--1.9, Definitions 2.3--2.8 and Theorem 2.10. Remark 1.8 identifies the absence of quantitative covering bounds. https://arxiv.org/abs/1110.5008.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 80 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.