arXiv subjects

Atticus Stonestrom

Publications and source records attributed to Atticus Stonestrom.

7 recordsLinked to original sources

arXiv2026

Some results on NIP groups and their Ellis groups

This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic form an ideal. Our hope is that the corresponding piecewise (strong) f-generic types can provide a substitute in arbitrary NIP groups for the (strong) f-generic types of definably amenable NIP groups, and in the rest of the paper we give several applications. Two of the applications deal with the Ellis group of an NIP group. Let $T$ be an NIP theory, $G$ a definable group, and $M$ a model. In our first result we show that the size of the Ellis group of $G(M)$ is bounded above by $2^{|T|}$, independent of the choice of $M$, giving a substantial step towards the question of whether the isomorphism type is independent of $M$. In our second result, inspired by a theorem of Hrushovski, we show that, if $T$ and $M$ are countable and the formulas of $T$ have uniformly bounded VC-codensity, then the Ellis group of $G(M)$ has `finite Archimedean rank', ie its connected component is profinite-by-Lie. A crucial tool for us in both results is the recent result of Chernikov-Gannon-Krupiński and Basso-Zucker that the $τ$-topology on the Ellis group is Hausdorff. Finally, we use our techniques to obtain a `local' result valid in arbitrary NIP theories, without the assumption of uniformly bounded VC-codensity: for any `bi-invariant' formula $ϕ(x,y)$, the group $G/G^{00}_ϕ$ has finite Archimedean rank. More precisely, if the VC-codensity of $ϕ(x,y)$ is at most $δ$, then $G/G^{00}_ϕ$ is an inverse limit of compact Lie groups of dimension at most $(4δ)^2$. This connects to, though is different than, a question of Hrushovski.

math.LO↗

arXiv2026

On non-abelian dp-minimal groups I: the torsion-free and distal cases

We give some results on dp-minimal groups. First we show that any torsion-free dp-minimal group is abelian; along the way show that any dp-minimal group admitting a principal f-generic type whose realizations are non-torsion is nilpotent-by-finite of class at most $2$. We then investigate the question of whether *every* dp-minimal group $G$ is nilpotent-by-finite. There are naturally two cases: either (1) $G$ admits a distal f-generic type or (2) $G$ admits a generically stable f-generic type. In this paper we resolve case (1). This follows from a more general structural analysis, in which, assuming that $G$ admits a distal f-generic type, we show that the quotient of $G$ by its FC-center can be naturally equipped with the structure of a valued group; we then use this valuation structure to show that indeed $G$ is nilpotent-by-finite. Case (2) of the question will be resolved in an upcoming joint paper with Eran Alouf and Frank Wagner.

math.LO↗

arXiv2026

On f-generic types in NIP groups

`Definable amenability' of a definable group is the model-theoretic analogue of amenability of a discrete group; precisely, a definable group is said to be definably amenable if it admits a translation-invariant finitely additive probability measure on its definable subsets. We prove a combinatorial characterization of definable amenability for groups definable in NIP theories. More specifically, given a group $G$, a subset $D\subseteq G$ is said to (left) `$G$-divide' if there is some natural number $k$ and an infinite sequence of elements $g_i\in G$ such that $g_{i_1}D\cap\dots\cap g_{i_k}D=\varnothing$ for all $i_1<\dots<i_k$. Our main result is that, if $G$ is a group definable in an NIP theory, and the union of two definable $G$-dividing subsets of $G$ still $G$-divides, then $G$ is definably amenable. It follows that $G$ is definably amenable if and only if $G$ admits a global non-$G$-dividing (or, equivalently, `f-generic') type. This answers a question of Chernikov and Simon and generalizes a theorem of Hrushovski and Pillay. As a quick application of the main result, we show that every dp-minimal group is definably amenable, which answers a question of Chernikov, Pillay, and Simon. Finally, we show that the appropriate analogue of the main result holds also for type-definable groups, so that, in an NIP theory, a type-definable group with a global f-generic type is definably amenable; this additionally gives the first correct proof of the analogous result, claimed by Hrushovski and Pillay, for type-definable groups with a global \textit{strongly} f-generic type.

math.LO↗

arXiv2026

An arithmetic algebraic regularity lemma

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field $\mathbf{F}$, and any definable group $(G,\cdot)$ in $\mathbf{F}$ and definable subset $D\subseteq G$, each of complexity at most $M$, there is a normal definable subgroup $H\leqslant G$, of index and complexity $O_M(1)$, such that the following holds: for any cosets $V,W$ of $H$, the bipartite graph $(V,W,xy^{-1}\in D)$ is $O_M(|\mathbf{F}|^{-1/2})$-quasirandom. Various analogous regularity conditions follow; for example, for any $g\in G$, the Fourier coefficient $||\widehat{1}_{H\cap Dg}(π)||_{\mathrm{op}}$ is $O_M(|\mathbf{F}|^{-1/8})$ for every non-trivial irreducible representation $π$ of $H$.

math.LO↗

arXiv2023

Forking and invariant measures in NIP theories

We give an example of an NIP theory $T$ in which there is a formula that does not fork over $\varnothing$ but has measure $0$ under any global $\varnothing$-invariant Keisler measure, and we show that this cannot occur if $T$ is also first-order amenable.

math.LO↗

arXiv2023

Product-free sets in approximate subgroups of distal groups

Recall that a subset $X$ of a group $G$ is 'product-free' if $X^2\cap X=\varnothing$, ie if $xy\notin X$ for all $x,y\in X$. Let $G$ be a group definable in a distal structure. We prove there are constants $c>0$ and $δ\in(0,1)$ such that every finite subset $X\subseteq G$ distinct from $\{1\}$ contains a product-free subset of size at least $δ|X|^{c+1}/|X^2|^c$. In particular, every finite $k$-approximate subgroup of $G$ distinct from $\{1\}$ contains a product-free subset of density at least $δ/k^c$. The proof is short, and follows quickly from Ruzsa calculus and an iterated application of Chernikov and Starchenko's distal regularity lemma.

math.CO↗

arXiv2021

Some model theory of $\operatorname{Th}(\mathbb{N},\cdot)$

'Skolem arithmetic' is the complete theory $T$ of the multiplicative monoid $(\mathbb{N},\cdot)$. We give a full characterization of the $\varnothing$-definable stably embedded sets of $T$, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that $T$ has weak elimination of imaginaries but not elimination of finite imaginaries.

math.LO↗