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Approximate subgroups with bounded VC-dimension

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Abstract

We combine the fundamental results of Breuillard, Green, and Tao (Publ Math Inst Hautes Études Sci 116:115–221, 2012) on the structure of approximate groups, together with “tame” arithmetic regularity methods based on work of the authors and Terry (J Eur Math Soc (JEMS) 24(2):583–621, 2022), to give a structure theorem for finite subsets A of arbitrary groups G where A has “small tripling” and bounded VC-dimension: Roughly speaking, up to a small error, A will be a union of a bounded number of translates of a coset nilprogression of bounded rank and step (see Theorem 2.1). We also prove a stronger result in the setting of bounded exponent (see Theorem 2.2). Our results extend recent work of Martin-Pizarro, Palacín, and Wolf (Selecta Math (N.S.) 27(4):Paper No. 53, 19,2021) on finite stable sets of small tripling.

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Notes

  1. We use [n] to denote \(\{1,\ldots ,n\}\).

  2. This acronym, which comes from model theory, stands for the “negation of the independence property”.

  3. The “complexity” of this Boolean combination can also be bounded in terms of the parameters d, k, r, and \(\epsilon \) only; see Remark 7.10.

  4. A group word is a term in the language of groups with a function symbol for inversion.

  5. To avoid ambiguity in words like “definable”, one can iterate the expansion by the symbols \(f^X_\phi \).

  6. In [39], coset progressions are not assumed to be symmetric, but it is easy to check that a Freiman 2-isomorphism preserving the identity also preserves symmetric sets.

  7. This is the abelian case of Ruzsa’s Covering Lemma (Lemma 7.2).

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Gabriel Conant

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Anand Pillay

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  1. Gabriel Conant
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Partially supported by NSF grants: DMS-1855503 (Conant); DMS-1665035, DMS-1790212 (Pillay).

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Conant, G., Pillay, A. Approximate subgroups with bounded VC-dimension. Math. Ann. 388, 1001–1043 (2024). https://doi.org/10.1007/s00208-022-02524-3

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