arXiv subjects
Publications and source records attributed to Daniel Palacín.
In this short note, we present a self-contained exposition of the supersimplicity of certain expansions of the additive group of the integers, such as adding a generic predicate (due to Chatzidakis and Pillay), a predicate for the square-free integers (due to Bhardwaj and Tran) or a predicate for the prime integers (due to Kaplan and Shelah, assuming Dickson's conjecture).
The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for definable groups in simple theories. In the last sections of this article, we apply our model-theoretic results to bound the number of initial points starting few arithmetic progression of length $3$ in the structure of the additive group of integers with a predicate for the prime integers, assuming Dickson's conjecture, or with a predicate for the square-free integers, as well as for asymptotic limits of finite fields. Our techniques yield similar results for the elements appearing as distances in skew-corners and for Sárközy's theorem on the distance of distinct elements being perfect squares.
We answer in the affirmative a conjecture of Berarducci, Peterzil and Pillay \cite{BPP10} for solvable groups, which is an o-minimal version of a particular case of Milnor's isomorphism conjecture \cite{jM83}. We prove that every abstract finite central extension of a definably connected solvable definable group in an o-minimal structure is equivalent to a definable (hence topological) finite central extension. The proof relies on an o-minimal adaptation of the higher inflation-restriction exact sequence due to Hochschild and Serre. As in \cite{jM83}, we also prove in o-minimal expansions of real closed fields that the conjecture reduces to definably simple groups.
The purpose of this simple note is to provide elementary model-theoretic proofs to some existing results on sumset phenomena and IP sets, motivated by Hrushovski's work on the stabilizer theorem.
We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil and Starchenko. In particular, we prove for definably compact groups that the space of closed points is homeomorphic to the space of infinitesimal types. We also prove that with the spectral topology the set of invariant types concentrated in a definably compact set is a normal spectral space whose closed points are the finitely satisfiable types. On the other hand, for arbitrary NIP structures we equip the set of invariant types with a new topology, called the {\em honest topology}. With this topology the set of invariant types is a normal spectral space whose closed points are the finitely satisfiable ones, and the natural retraction from invariant types onto finitely satisfiable types coincides with Simon's $F_M$ retraction.
We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. We also obtain a model-theoretic proof of Roth's theorem on the existence of arithmetic progressions of length $3$ for subsets of positive density in suitable definably amenable groups, such as countable amenable abelian groups without involutions and ultraproducts of finite abelian groups of odd order.
We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgrebraic functions representing Boolean combinations of d-semialgebraic sets.
We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we also construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic $0$ give rise to triangular rings without the CBP. This also applies to Baudisch's $2$-step nilpotent Lie algebras, which yields the existence of a $2$-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.
The main goal of the paper is to present a general model theoretic framework to understand a result of Shalev on probabilistically finite nilpotent groups. We prove that a suitable group where the equation $[x_1,\ldots,x_k]=1$ holds on a wide set, in a model theoretic sense, is an extension of a nilpotent group of class less than $k$ by a uniformly locally finite group. In particular, this result applies to amenable groups, as well as to suitable model-theoretic families of definable groups such as groups in simple theories and groups with finitely satisfiable generics.
Every finite non-abelian group of order $n$ has a non-central element whose centralizer has order exceeding $n^{1/3}$. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.
A subset of a group is said to be product-free if it does not contain three elements satisfying the equation $xy=z$. We give a negative answer to a question of Babai and Sós on the existence of large product-free sets by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have a large product-free subset, as well as a model theoretic account of a result of Nikolov and Pyber on triple products.
An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik-Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups $A$ such that there is only finitely many primes $p$ such that the group $A/pA$ is infinite and for every prime $p$, there is only finitely many natural numbers $n$ such that $(p^nA)[p]/(p^{n+1}A)[p]$ is infinite. Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.
A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.
It is proven that an infinite finitely generated group cannot be elementarily equivalent to an ultraproduct of finite groups of a given Prüfer rank. Furthermore, it is shown that an infinite finitely generated group of finite Prüfer rank is not pseudofinite.
In this paper we shall prove that any $2$-transitive finitely homogeneous structure with a supersimple theory satisfying a generalized amalgamation property is a random structure. In particular, this adapts a result of Koponen for binary homogeneous structures to arbitrary ones without binary relations. Furthermore, we point out a relation between generalized amalgamation, triviality and quantifier elimination in simple theories.
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.
In \cite{HPP}, Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that $T$ has the canonical base property in a strong form, " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than \cite{HPP}, and prove its {\em equivalence} with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that $1$-based groups are rigid.