Model Theory and Applications to Groups and Combinatorics / Théorie des modèles et applications en théorie des groupes et en combinatoire

Collection Model Theory and Applications to Groups and Combinatorics / Théorie des modèles et applications en théorie des groupes et en combinatoire

Organizer(s) Ben Yaacov, Itaï ; Hempel, Nadja ; Hils, Martin ; Zou, Tingxiang

Date(s) 30/09/2024 - 04/10/2024

linked URL https://conferences.cirm-math.fr/3112.html

00:00:00 / 00:00:00

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Almost periodicity from a model-theoretic perspective

By Amador Martin-Pizarro

Roth's theorem states that a subset $A$ of $\{1, \ldots, N}$ of positive density contains a positive $N^2$-proportion of (non-trivial) three arithmetic progressions, given by pairs $(a, d)$ with $d \neq 0$ such that $a, a+d, a+2 d$ all lie in $A$. In recent breakthrough work by Kelley and Meka, the known bounds have been improved drastically. One of the core ingredients of the their proof is a version of the almost periodicity result due to Croot and Sisask. The latter has been obtained in a non-quantitative form by Conant and Pillay for amenable groups using continuous logic. In joint work with Daniel Palacín, we will present a model-theoretic version (in classical first-order logic) of the almost-periodicity result for a general group equipped with a Keisler measure under some mild assumptions and show how to use this result to obtain a non-quantitative proof of Roth's result. One of the main ideas of the proof is an adaptation of a result of Pillay, Scanlon and Wagner on the behaviour of generic types in a definable group in a simple theory.

Information about the video

Date of recording 03/10/2024
Date of publication 28/10/2024
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Luca Recanzone
Format MP4

Citation data

DOI 10.24350/CIRM.V.20252703
Cite this video Martin-Pizarro, Amador (03/10/2024). Almost periodicity from a model-theoretic perspective. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20252703
URL https://dx.doi.org/10.24350/CIRM.V.20252703

Domain(s)

Logic

Bibliography

MARTIN-PIZARRO, Amador et PALACÍN, Daniel. Complete type amalgamation for nonstandard finite groups. Model Theory, 2024, vol. 3, no 1, p. 1-37. - https://doi.org/10.2140/mt.2024.3.1

MSC codes

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