Quantitative structure of stable sets in arbitrary finite groups

  title={Quantitative structure of stable sets in arbitrary finite groups},
  author={Gabriel Conant},
  journal={Proceedings of the American Mathematical Society},
  • G. Conant
  • Published 6 April 2020
  • Mathematics
  • Proceedings of the American Mathematical Society
We show that a $k$-stable set in a finite group can be approximated, up to given error $\epsilon>0$, by left cosets of a subgroup of index $\epsilon^{\text{-}O_k(1)}$. This improves the bound in a similar result of Terry and Wolf on stable arithmetic regularity in finite abelian groups, and leads to a quantitative account of work of the author, Pillay, and Terry on stable sets in arbitrary finite groups. We also prove an analogous result for finite stable sets of small tripling in arbitrary… 
2 Citations
Approximate subgroups with bounded VC-dimension
We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to
A model-theoretic note on the Freiman–Ruzsa theorem
A non-quantitative version of the Freiman-Ruzsa theorem is obtained for finite stable sets with small tripling in arbitrary groups, as well as for (finite) weakly normal subsets in abelian groups.


Structure and regularity for subsets of groups with finite VC-dimension
Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen
Convolutions of sets with bounded VC-dimension are uniformly continuous
We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ A \cap(xA) : x \in A\cdot A^{-1} \}$. We show that if a finite subset
Stability in a group
  • G. Conant
  • Mathematics
    Groups, Geometry, and Dynamics
  • 2021
We develop local stable group theory directly from topological dynamics, and extend the main results in this subject to the setting of stability "in a model". Specifically, given a group $G$, we
On finite sets of small tripling or small alternation in arbitrary groups
  • G. Conant
  • Mathematics
    Combinatorics, Probability and Computing
  • 2020
A qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularityLemma for sets of bounded VC-dimension in finite groups of bounded exponent are obtained.
Stable arithmetic regularity in the finite field model
  • C. TerryJ. Wolf
  • Mathematics, Computer Science
    Bulletin of the London Mathematical Society
  • 2018
An arithmetic regularity lemma is proved for k ‐stable subsets A⊆Fpn in which the bound on the codimension of the subspace is a polynomial (depending on k ) in the degree of uniformity, and in which there are no non‐uniform cosets.
Multiplicative structure in stable expansions of the group of integers
We define two families of expansions of $(\mathbb{Z},+,0)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega$. The first family consists of expansions
Sum-free sets in abelian groups
AbstractWe show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is $$\left( {2^{\nu (G)} - 1} \right)2^{\left| G \right|/2} + O\left(
Quantitative structure of stable sets in finite abelian groups
  • C. TerryJ. Wolf
  • Mathematics
    Transactions of the American Mathematical Society
  • 2020
We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A
A group version of stable regularity
Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index
A Szemerédi-type regularity lemma in abelian groups, with applications
Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the