The rectifiability threshold in abelian groups


For any abelian group G and integer t ≥ 2 we determine precisely the smallest possible size of a non-t-rectifiable subset of G. Specifically, assuming that G is not torsion-free, denote by p the smallest order of a non-zero element of G. We show that if a subset SG satisfies |S| ≤ ⌌log t p⌍, then S is t-isomorphic (in the sense of Freiman) to a set of integers; on the other hand, we present an example of a subset SG with |S| = ⌌log t p⌍ + 1 which is not t-isomorphic to a set of integers.

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Correspondence to Vsevolod F. Lev.

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Lev, V.F. The rectifiability threshold in abelian groups. Combinatorica 28, 491–497 (2008).

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Mathematics Subject Classification (2000)

  • 11P70
  • 11B75