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Quasirandomness of definable subsets of definable groups in finite fields

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Authors : Pillay, Anand (Author of the conference)
CIRM (Publisher )

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Abstract : We give an arithmetic version of Tao's algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs $(G, D)$ where $G$ is a group and $D$ an arbitrary subset, all uniformly definable in finite fields. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph $(G, G, E)$ where $E(x, y)$ is $x y^{-1} \in D$, witnessed by a the decomposition of $G$ into cosets of a uniformly definable small index normal subgroup $H$ of $G$.

Keywords : Arithmetic regularity; finite fields

MSC Codes :
03C45 - Classification theory, stability and related concepts [See also 03C48]
05C75 - Structural characterization of types of graphs
11B30 - Arithmetic combinatorics; higher degree uniformity

Additional resources :
https://www.cirm-math.fr/RepOrga/3112/Slides/Anand-Pillay.pdf

    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 : Algebraic Geometry ; Logic
    Mathematical Area(s) : Algebra ; Algebraic & Complex Geometry ; Logic and Foundations
    Format : MP4 (.mp4) - HD
    Video Time : 00:53:59
    Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-10-03_Pillay.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.20252803
Cite this video as: Pillay, Anand (2024). Quasirandomness of definable subsets of definable groups in finite fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20252803
URI : http://dx.doi.org/10.24350/CIRM.V.20252803

See Also

Bibliography

  • TAO, Terence. Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets. arXiv preprint arXiv:1211.2894, 2012. - https://arxiv.org/abs/1211.2894

  • GOWERS, W. T. Quasirandom groups, eprint. arXiv, 2007, vol. 710. - https://arxiv.org/abs/0710.3877

  • GREEN, Ben. A Szemerédi-type regularity lemma in abelian groups, with applications. Geometric & Functional Analysis GAFA, 2005, vol. 15, no 2, p. 340-376.
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