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Almost periodicity from a model-theoretic perspective

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Authors : Martin-Pizarro, Amador (Author of the conference)
CIRM (Publisher )

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Abstract : 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.

Keywords : Model theory; additive combinatorics; arithmetic progressions

MSC Codes :
03C45 - Classification theory, stability and related concepts [See also 03C48]
11B30 - Arithmetic combinatorics; higher degree uniformity

    Information on the Video

    Film maker : Recanzone, Luca
    Language : English
    Available date : 28/10/2024
    Conference Date : 03/10/2024
    Subseries : Research talks
    arXiv category : Logic
    Mathematical Area(s) : Combinatorics ; Logic and Foundations
    Format : MP4 (.mp4) - HD
    Video Time : 00:52:27
    Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-10-03_martin_pizarro.mp4

Information on the Event

Event Title : Model Theory and Applications to Groups and Combinatorics / Théorie des modèles et applications en théorie des groupes et en combinatoire
Event Organizers : Ben Yaacov, Itaï ; Hempel, Nadja ; Hils, Martin ; Zou, Tingxiang
Dates : 30/09/2024 - 04/10/2024
Event Year : 2024
Event URL : https://conferences.cirm-math.fr/3112.html

Citation Data

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

See Also

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



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