arXiv subjects

Ehud Hrushovski

Publications and source records attributed to Ehud Hrushovski.

At least 19 recordsLinked to original sources

arXiv2024

Globally valued fields: foundations

We present foundations for globally valued fields, a class of enriched fields capturing some aspects of the geometry of global fields, based on the product formula. We show that this class is axiomatizable in the framework of unbounded continuous logic, opening the way to decidability and transfer theorems. We provide a dictionary between various data defining such extra structure: syntactic, Arakelov or cycle-theoretic, and measure theoretic. In particular we obtain a representation theorem relating globally valued fields and adelic curves defined by Chen and Moriwaki.

math.LO↗

arXiv2024

Approximate equivalence relations

We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover's normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim-Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov-Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an 'approximately unique' probability logic 1-type, we obtain a structure theorem generalizing the `Lie model' theorem for approximate subgroups. The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. For NIP theories, pursuing a question of Pillay's, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets.

math.LO↗

arXiv2024

Model Equivalences

We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of definable MERs, along with the first steps of a classification theory for them. We characterize the special classes of definable MERs associated with preservation of formulas, either in classical first order logic or in continuous logic, and uncover an intrinsic role for the latter. We bring out a nontrivial relationship with interpretations (imaginary sorts), leading to a wider hierarchy of classes related to the preservation of reducts. We give results about the relationship between these classes, both for general theories and for theories satisfying additional model-theoretic properties, such as stability.

math.LO↗

arXiv2024

Lang-Weil Type Estimates in Finite Difference Fields

We prove a uniform estimate of the number of points for difference algebraic varieties in finite difference fields in the spirit of Lang-Weil. More precisely, we give uniform lower and upper bounds for the number of rational points of a difference variety in terms of its transformal dimension. As a main technical ingredient, we prove an equidimensionality result for Frobenius reductions of difference varieties.

math.NT↗

arXiv2023

Imaginaries, products and the adele ring

We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of finite subsets of $I$. As a special case, using the Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the imaginary sorts of the ring of rational adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning from monadic second order imaginaries to first-order reducts, and for proving a certain ``one-way'' model-theoretic orthogonality within the adelic imaginaries.

math.LO↗

arXiv2023

Compactifications of pseudofinite and pseudo-amenable groups

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.

math.LO↗

arXiv2023

The Theory of the Entire Algebraic Functions

Let A be the integral closure of the ring of polynomials CC[t], within the field of algebraic functions in one variable. We show that A interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory (see Prestel-Schmid and van den Dries-A. Macintyre).

math.LO↗

arXiv2023

Embedded Finite Models Beyond Restricted Quantifier Collapse

We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as an interpreted vocabulary over an infinite domain. This formalism was denoted embedded finite model theory in the past. It is clear that the expressiveness and evaluating complexity of formulas of this type depends heavily on the infinite structure. If we embed in a wild structure like the integers with additive and multiplicative arithmetic, logic is extremely expressive and formulas are impossible to evaluate. On the other hand, for some well-known decidable structures, the expressiveness and evaluating complexity are similar to the situation without the additional infrastructure. The latter phenomenon was formalized via the notion of ``Restricted Quantifier Collapse'': adding quantification over the infinite structure does not add expressiveness. Beyond these two extremes little was known. In this work we show that the possibilities for expressiveness and complexity are much wider. We show that we can get almost any possible complexity of evaluation while staying within a decidable structure. We also show that in some decidable structures, there is a disconnect between expressiveness of the logic and complexity, in that we cannot eliminate quantification over the structure, but this is not due to an ability to embed complex relational computation in the logic. We show failure of collapse for the theory of finite fields and the related theory of pseudo-finite fields, which will involve coding computation in the logic. As a by-product of this, we establish new lower-bounds for the complexity of decision procedures for several decidable theories of fields, including the theory of finite fields. In the process of investigating this landscape, we investigate several weakenings of collapse.

cs.LO↗

arXiv2022

Globally valued function fields: existential closure

These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof that {\em the canonical GVF structure on $k(t)^{alg}$ is existentially closed}. This can be read as saying that a variety {\em with a distinguished curve class} is a good approximation for a formula in the language of GVFs, in the same way that a variety is close to a formula for the theory ACF of algebraically closed fields.

math.LO↗

arXiv2022

Specialization of Difference Equations and High Frobenius Powers

We study valued fields equipped with an automorphism $\sigma$ which is locally infinitely contracting in the sense that $\alpha\ll\sigma\alpha$ for all $0<\alpha\in\Gamma$. We show that various notions of valuation theory, such as Henselian and strictly Henselian hulls, admit meaningful transformal analogues. We prove canonical amalgamation results, and exhibit the way that transformal wild ramification is controlled by torsors over generalized vector groups. Model theoretically, we determine the model companion: it is decidable, admits a simple axiomatization, and enjoys elimination of quantifiers up to algebraically bounded quantifiers. The model companion is shown to agree with the limit theory of the Frobenius action on an algebraically closed and nontrivially valued field. This opens the way to a motivic intersection theory for difference varieties that was previously available only in characteristic zero. As a first consequence, the class of algebraically closed valued fields equipped with a distinguished Frobenius $x\mapsto x^{q}$ is decidable, uniformly in $q$.

math.LO↗

arXiv2022

Tropical functions on a skeleton

We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework of stable completions of algebraic varieties, a model-theoretic version of Berkovich analytifications, for which we prove a similar result, of which the former one is a consequence.

math.AG↗

arXiv2022

Piecewise Interpretable Hilbert Spaces

We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a first-order theory. Under a new finiteness hypothesis that we call {\em scatteredness} we prove that $H$ is a direct sum of {\em asymptotically free} components, where short-range interactions are controlled by algebraic closure and long-range interactions vanish. Examples include $L^2$-spaces relative to Macpherson-Steinhorn definable measures; $L^2$ spaces relative to the Haar measure of the absolute Galois groups; irreducible unitary representations of $p$-adic Lie groups; and unitary representations of the automorphism group of an $ω$-categorical theory. In the last case, our main result specialises to a theorem of Tsankov. New methods are required, making essential use of local stability theory in continuous logic.

math.LO↗

arXiv2022

Invariant measures in simple and in small theories

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.

math.LO↗

arXiv2022

The Elementary Theory of the Frobenius Automorphisms

A Frobenius difference field is an algebraically closed field of characteristic $p>0$, enriched with a symbol for $x \mapsto x^{p^m}$. We study a sentence or formula in the language of fields with a distinguished automorphism, interpreted in Frobenius difference fields with $p$ or $m$ tending to infinity. In particular, a decision procedure is found to determine when a sentence is true in almost every Frobenius difference field. This generalizes Cebotarev's density theorem and Weil's Riemann hypothesis for curves (both in qualitative versions), but hinges on a result going slightly beyond the latter. The setting for the proof is the geometry of difference varieties of transformal dimension zero; these generalize algebraic varieties, and are shown to have a rich structure, only partly explicated here. Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations.

math.LO↗

arXiv2022

Beyond the Lascar Group

We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {\em core} $\mathcal{J}$ associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $\mathcal{G}$. The automorphism groups of models of the theory are related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em quasi-homomorphism}, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of $SL_n({\mathbb{R}})$ or $SL_n({\mathbb{Q}}_p)$.

math.LO↗

arXiv2022

Definability patterns and their symmetries

We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the Kim-Pillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Hausdorff part J; the general Lascar group can be read off this compact structure. This gives concrete form to results of Krupiński, Newelski, Pillay, Rzepecki and Simon, who used topological dynamics applied to large models to show the existence of compact groups mapping onto the Lascar group. In an appendix, we show that a construction analogous to the above but using infinitary patterns recovers the Ellis group of \cite{kns}, and use this to sharpen the cardinality bound for their Ellis group from $\beth_5$ to $\beth_3$, showing the latter is optimal. There is also a close connection to another school of topological dynamics, set theory and model theory, centered around the Kechris-Pestov-Todor\v cević correspondence. We define the Ramsey property for a first order theory, and show - as a simple application of the construction applied to an auxiliary theory - that any theory admits a canonical minimal Ramsey expansion. This was envisaged and proved for certain Fraissé classes, first by Kechris-Pestov-Todor\v cević for expansions by orderings, then by Melleray, Nguyen Van Thé, Tsankov and Zucker for more general expansions.

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↗

arXiv2021

Ax's theorem with an additive character

Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann hypothesis for curves over finite fields is taken by the `Weil bound' on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions $\Ff_p^n \to \Cc$ obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $Ψ_p(ξ)$ at certain algebraic functions $ξ$ of the variables, where $Ψ(n \mod p) = exp(2 πi n/p)$ is the standard additive character.

math.LO↗