arXiv subjects

Krzysztof Krupiński

Publications and source records attributed to Krzysztof Krupiński.

17 recordsLinked to original sources

arXiv2026

Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture

We study idempotent measures and the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient to develop stable group theory localizing on a generically transitive type. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Finally, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup. In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups.

math.LO↗

arXiv2026

Ramsey theory and topological dynamics of 0-dimensional flows

We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.

math.DS↗

arXiv2026

Haar decompression and amenability of Ellis flows

Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $τ$-topology, the inclusion $K\hookrightarrow E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $μ_K$ on $E(X,G)$, called its Haar decompression. Haar decompression gives an exact ergodicity description. For an arbitrary flow $(X,G)$, amenability of its Ellis flow is equivalent to hereditary amenability of all finite powers $X^n$. In the tame setting, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. Moreover, for the minimal left ideal $\mathcal{M}$ in $E(X,G)$ containing $K$, $μ_K$ is $G$-invariant if and only if $(\mathcal{M},G)$ is amenable; when this holds, $μ_K$ is the unique ergodic measure on $\mathcal{M}$ and $\mathcal{M}=\overline{K}$. Then, we conclude that the ergodic measures of $E(X,G)$ are precisely the Haar decompressions associated with amenable minimal left ideals. We also prove that every minimal tame amenable flow is uniquely ergodic and that every ergodic invariant measure on a tame flow has minimal support. Consequently, every ergodic measure on a tame ambit arises by evaluating a suitable Haar decompression.

math.DS↗

arXiv2026

Generalized locally compact models for approximate groups

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one obtained by Hrushovski in ``Beyond the Lascar group'', and it uses only basic model theory (mostly spaces of types and realizations of types). The main tools are Ellis groups from topological dynamics considered for suitable spaces of types. However, we need to redevelop some basic theory of topological dynamics for suitable ``locally compact flows'' in place of (compact) flows. We also prove that the generalized definable locally compact model which we constructed is universal in an appropriate category. We note that the main result yields structural information on definable generic subsets of definable groups, with a more precise structural result for generics in the universal cover of $\textrm{SL}_2(\mathbb{R})$.

math.LO↗

arXiv2026

On the structure of approximate rings

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate subrings. Our aim is to develop a general framework for the sum-product phenomenon that applies uniformly across arbitrary rings. The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication. Another application of the main structure theorem is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24]. This existence theorem extends beyond the finite (and pseudofinite) setting. To illustrate the scope of the method, we also establish a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing a classical sum-product result of Meyer.

math.RA↗

arXiv2025

On first order amenability

We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $ω$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by $\emptyset$-definable global Keisler measures (which is for example the case for amenable $ω$-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.

math.LO↗

arXiv2025

Convolution semigroups for automorphism dynamics

Initially motivated by Hrushovski's paper on definability patterns, we obtain homeomorphisms between Ellis semigroups related to natural actions of the automorphism groups of first order structures and certain collections of types and Keisler measures. Thus, we can transfer the semigroup operation from these Ellis semigroups to the corresponding collections of types and Keisler measures. By generalizing this transferred product, we obtain a new convolution operation for invariant types and measures in arbitrary first-order theories. We develop its general theory and prove several correspondence theorems between idempotent measures and closed subgroups of the automorphism group of a sufficiently large (so-called monster) model with respect to the relatively definable topology. Via the affine sort construction, we demonstrate that this new notion of convolution encodes the standard definable convolution operation over definable groups.

math.LO↗

arXiv2025

Maximal WAP and tame quotients of type spaces

We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\mathfrak{C})$, where $\mathfrak{C}$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\emptyset$-type-definable set, and $S_X(\mathfrak{C})$ is the space of complete types over $\mathfrak{C}$ concentrated on $X$. Namely, let $F_{\textrm{WAP}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ is WAP, and let $F_{\textrm{Tame}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}} )$ is tame. We show good behaviour of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ under changing the monster model $\mathfrak{C}$. Namely, we prove that if $\mathfrak{C}'\succ \mathfrak{C}$ is a bigger monster model, $F'_{\textrm{WAP}}$ and $F'_{\textrm{Tame}}$ are the counterparts of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ computed for $\mathfrak{C}'$, and $r\colon S_X(\mathfrak{C}')\to S_X(\mathfrak{C})$ is the restriction map, then $r[F'_{\textrm{WAP}}]=F_{\textrm{WAP}}$ and $r[F'_{\textrm{Tame}}]=F_{\textrm{Tame}}$. Using these results, we show that the Ellis (or ideal) groups of $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ and $(aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}})$ do not depend on the choice of the monster model $\mathfrak{C}$.

math.LO↗

arXiv2023

Maximal stable quotients of invariant types in NIP theories

For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$, and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we prove that there exists a finest relatively type-definable over a small set of parameters from $\mathfrak{C}$ equivalence relation on the set of realizations of $p$ which has stable quotient. This is a counterpart for equivalence relations of the main result of the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay which shows the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in the paper "On first order amenability" by E. Hrushovski, K. Krupinski, and A. Pillay.

math.LO↗

arXiv2023

Locally compact models for approximate rings

By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. We prove that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component. The above theorem can be seen as a general structural result about approximate subrings: every approximate subring $X$ can be recovered up to additive commensurability as the preimage by a locally compact model $f \colon \langle X \rangle \to S$ of any relatively compact neighborhood of $0$ in $S$. It also leads to more precise structural or even classification results. For example, we deduce that every [definable] approximate subring $X$ of a ring of positive characteristic is additively commensurable with a [definable] subring contained in $4X + X \cdot 4X$. This implies that for any given $K,L \in \mathbb{N}$ there exists $C(K,L)$ such that every $K$-approximate subring $X$ (i.e. $K$ additive translates of $X$ cover $X \cdot X \cup (X+X)$) of a ring of positive characteristic $\leq L$ is additively $C(K,L)$-commensurable with a subring contained in $4X + X \cdot 4X$. We also deduce a classification of finite approximate subrings of rings without zero divisors: for every $K \in \mathbb{N}$ there exists $N(K) \in \mathbb{N}$ such that for every finite $K$-approximate subring $X$ of a ring without zero divisors either $|X| <N(K)$ or $4X + X \cdot 4X$ is a subring which is additively $K^{11}$-commensurable with $X$.

math.LO↗

arXiv2022

Generating ideals by additive subgroups of rings

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let $R$ be any ring equipped with an arbitrary additional first order structure, and $A$ a set of parameters. We show that whenever $H$ is an $A$-definable, finite index subgroup of $(R,+)$, then $H+RH$ contains an $A$-definable, two-sided ideal of finite index. As a corollary, we positively answer Question 3.9 of [Bohr compactifications of groups and rings, J. Gismatullin, G. Jagiella and K. Krupiński]: if $R$ is unital, then $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$, where $\bar R \succ R$ is a sufficiently saturated elementary extension of $R$, and $(\bar R,+)^{00}_A$ [resp. $\bar R^{00}_A$] is the smallest $A$-type-definable, bounded index additive subgroup [resp. ideal] of $\bar R$. This implies that $\bar R^{00}_A=\bar R^{000}_A$, where $\bar R^{000}_A$ is the smallest invariant over $A$, bounded index ideal of $\bar R$. If $R$ is of finite characteristic (not necessarily unital), we get a sharper result: $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$. We obtain similar results for finitely generated (not necessarily unital) rings and for topological rings. The above results imply that the simplified descriptions of the definable (so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of the aforementioned paper are valid for all unital rings. We analyze many examples, where we compute the number of steps needed to generate a group by $(\bar R \cup \{1\}) \cdot (\bar R,+)^{00}_A$ and study related aspects, showing "optimality" of some of our main results and answering some natural questions.

math.LO↗

arXiv2021

Amenability, connected components, and definable actions

We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjectures from [24]. We introduce $\bigvee$-definable group topologies on a given $\emptyset$-definable group $G$ (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. We study the relationship between definability of an action of a definable group on a compact space, weakly almost periodic actions, and stability. We conclude that for any group $G$ definable in a sufficiently saturated structure, every definable action of $G$ on a compact space supports a $G$-invariant probability measure. This gives negative solutions to some questions and conjectures from [22] and [24]. We give an example of a $\emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $\mathbb{F}_2 \times \mathbb{Z}$ in a suitable language for which the $\bigvee$-definable group $H:=\langle X \rangle$ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact "model" exists for each approximate subgroup does not work in general.

math.LO↗

arXiv2018

Galois groups as quotients of Polish groups

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_σ$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $\emptyset$, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the aforementioned paper about bounded quotients of type-definable subgroups of definable groups.

math.LO↗

arXiv2018

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g. we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over $\emptyset$ is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $\mathfrak{C}$ be a monster model of a countable theory, $p \in S(\emptyset)$, and $E$ be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on $p(\mathfrak{C})$. Then, exactly one of the following holds: (1) $E$ is relatively definable (on $p(\mathfrak{C})$), smooth, and has finitely many classes, (2) $E$ is not relatively definable, but it is type-definable, smooth, and has $2^{\aleph_0}$ classes, (3) $E$ is not type definable and not smooth, and has $2^{\aleph_0}$ classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.

math.LO↗

arXiv2015

Smoothness of bounded invariant equivalence relations

We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to get the conclusion.

math.LO↗

arXiv2013

Locally finite profinite rings

We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson radical of any locally finite profinite ring is nil of finite nilexponent. Our results apply to the context of small compact $G$-rings, where we also obtain a description of possible actions of $G$ on the underlying ring.

math.LO↗

arXiv2012

Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.

math.LO↗