Hostname: page-component-5cf477f64f-fcbfl Total loading time: 0 Render date: 2025-03-28T23:11:24.145Z Has data issue: false hasContentIssue false
  • English
  • Français

DIMENSION AND MEASURE IN PSEUDOFINITE H-STRUCTURES

Part of: Model theory

Published online by Cambridge University Press:  18 March 2025

ALEXANDER BERENSTEIN Open the ORCID record for ALEXANDER BERENSTEIN [Opens in a new window] ,
DARÍO GARCÍA Open the ORCID record for DARÍO GARCÍA [Opens in a new window]  and
TINGXIANG ZOU Open the ORCID record for TINGXIANG ZOU [Opens in a new window]
ALEXANDER BERENSTEIN*
Affiliation:
DEPARTAMENTO DE MATEMÁTICAS UNIVERSIDAD DE LOS ANDES CARRERA 1 NO. 18A-12, EDIFICIO H, BOGOTÁ 111711, COLOMBIA E-mail: [email protected]
DARÍO GARCÍA
Affiliation:
DEPARTAMENTO DE MATEMÁTICAS UNIVERSIDAD DE LOS ANDES CARRERA 1 NO. 18A-12, EDIFICIO H, BOGOTÁ 111711, COLOMBIA E-mail: [email protected]
TINGXIANG ZOU
Affiliation:
UNIVERSITÉ CLAUDE BERNARD - LYON 1 CNRS 5208 INSTITUT CAMILLE JORDAN 43 BOULEVARD. DU 11 NOVEMBRE 1918 F.69622 VILLEURBANNE CEDEX, FRANCE Current address: FACHBEREICH MATHEMATIK UND INFORMATIK DER UNIVERSITÄT MÜNSTER ORLÉANS-RING 10, 48149 MÜNSTER GERMANY E-mail: [email protected]
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study H-structures associated with $SU$-rank 1 measurable structures. We prove that the $SU$-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.

Keywords

supersimple theoriesH-structuresmeasurable structuresasymptotic classespseudofinite dimension

MSC classification

Primary: 03C45: Classification theory, stability and related concepts 03C20: Ultraproducts and related constructions
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 37
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anscombe, S., Macpherson, D., Steinhorn, C., and Wolf, D., Multidimensional asymptotic classes and generalised measurable structures, in preparation, 2024, https://arxiv.org/pdf/2408.00102.Google Scholar
Berenstein, A. and Vassiliev, E., On lovely pairs of geometric structures . Annals of Pure and Applied Logic, vol. 161 (2010), no. 7, pp. 866–878.CrossRefGoogle Scholar
Berenstein, A. and Vassiliev, E., Generic trivializations of geometric theories . Mathematical Logic Quarterly, vol. 60 (2014), nos. 4–5, pp. 243–371.CrossRefGoogle Scholar
Berenstein, A. and Vassiliev, E., Geometric structures with a dense independent subset . Selecta Mathematica, vol. 22 (2016), no. 1, pp. 191–225.CrossRefGoogle Scholar
Carmona, J. F., Forking geometry on theories with an independent predicate . Archive for Mathematical Logic, vol. 54 (2015), pp. 247–255.CrossRefGoogle Scholar
Chatzidakis, Z., Model theory of difference fields , The Notre Dame Lectures (Cholak, P., editor), Lecture Notes in Logic, Cambridge University Press, Cambridge, 2005, pp. 45–94.Google Scholar
Chatzidakis, Z. and Pillay, A., Generic structures and simple theories . Annals of Pure and Applied Logic, vol. 95 (1998), nos. 1–3, pp. 71–92.CrossRefGoogle Scholar
Chernikov, A. and Simon, P., Definably amenable NIP groups . Journal of the American Mathematical Society, vol. 31 (2018), no. 3, pp. 609–641.CrossRefGoogle Scholar
Elwes, R. and Macpherson, D., A survey of asymptotic classes and measurable structures , Model Theory with Applications to Algebra and Analysis, vol. 2, (Chatzidakis, Z., Macpherson, D., Pillay, A. and Wilkie, A. editors), London Mathematical Society Lecture Note Series, 350, Cambridge University Press, Cambridge, 2008, pp. 125–159.CrossRefGoogle Scholar
Macpherson, D. and Steinhorn, C., One-dimensional asymptotic classes of finite structures . Transactions of the American Mathematical Society, vol. 360 (2007), pp. 411–448.CrossRefGoogle Scholar
Pillay, A., Strongly minimal pseudofinite structures, preprint, 2014, https://arxiv.org/pdf/1411.5008.pdf.Google Scholar
Vassiliev, E., Generic pairs of $SU$ -rank 1 structures . Annals of Pure and Applied Logic, vol. 120 (2003), pp. 103–149.CrossRefGoogle Scholar
Wolf, D., Multidimensional asymptotic classes of finite structures, Ph.D. thesis, University of Leeds, 2016.Google Scholar
Zou, T., Pseudofinite $H$ -structures and groups definable in supersimple $H$ -structures . Journal of Symbolic Logic, vol. 84 (2019), no. 3), pp. 937–956.CrossRefGoogle Scholar
Zou, T., Pseudofinite difference fields and counting dimensions . Journal of Mathematical Logic, vol. 21 (2021), p. 2050022.CrossRefGoogle Scholar