JournalsjemsVol. 24, No. 2pp. 583–621

Structure and regularity for subsets of groups with finite VC-dimension

  • Gabriel Conant

    University of Cambridge, UK
  • Anand Pillay

    University of Notre Dame, USA
  • Caroline Terry

    University of Chicago, USA
Structure and regularity for subsets of groups with finite VC-dimension cover
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Abstract

Suppose GG is a finite group and AGA\subseteq G is such that {gA:gG}\{gA:g\in G\} has VC-dimension strictly less than kk. We find algebraically well-structured sets in GG which, up to a chosen ϵ>0\epsilon>0, describe the structure of AA and behave regularly with respect to translates of AA. For the subclass of groups with uniformly fixed finite exponent rr, these algebraic objects are normal subgroups with index bounded in terms of kk, rr, and ϵ\epsilon. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model-theoretic methods related to the work of Breuillard, Green, and Tao [8] and Hrushovski [28] on approximate groups, as well as a result of Alekseev, Glebskiĭ, and Gordon [1] on approximate homomorphisms.

Cite this article

Gabriel Conant, Anand Pillay, Caroline Terry, Structure and regularity for subsets of groups with finite VC-dimension. J. Eur. Math. Soc. 24 (2022), no. 2, pp. 583–621

DOI 10.4171/JEMS/1111