A Step Beyond Kemperman's Structure Theorem

Grynkiewicz, David J.

doi:10.1112/s0025579300000966

Cited by 20 publications

(31 citation statements)

References 33 publications

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“…While there are many such results approximating the structure of A and B, particularly in special groups, there are very few that fully characterize the possibilities, especially for an unrestricted abelian group G. One such result is due to Kemperman [11,Chapter 9] [13] [14], who gave a full characterization of when |A + B| < |A| + |B|. This was later extended to a characterization of when |A + B| ≤ |A| + |B| in [9], generalizing partial work achieved in [12]. They include some unwieldy possibilities, particularly when |A + B| is large in comparison to |G|, leading us to defer the relevant details until Section 2.…”

Section: 2mentioning

confidence: 99%

Iterated sumsets and subsequence sums

Grynkiewicz

2018

Journal of Combinatorial Theory, Series A

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Let G ∼ = Z/m1Z × . . . × Z/mrZ be a finite abelian group with 1 < m1 | . . . | mr = exp(G). The Kemperman Structure Theorem characterizes all subsets A, B ⊆ G satisfying |A + B| < |A| + |B| and has been extended to cover the case when |A + B| ≤ |A| + |B|. Utilizing these results, we provide a precise structural description of all finite subsets A ⊆ G with |nA| ≤ (|A| + 1)n − 3 when n ≥ 3 (also when G is infinite), in which case many of the pathological possibilities from the case n = 2 vanish, particularly for large n ≥ exp(G) − 1. The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence S of terms from G having length |S| ≥ 2|G| − 1 must either have every element of G representable as a sum of |G|-terms from S or else have all but |G/H| − 2 of its terms lying in a common H-coset for some H ≤ G. We show that the much weaker hypothesis |S| ≥ |G| + exp(G) suffices to obtain a nearly identical conclusion, where for the case H is trivial we must allow all but |G/H|−1 terms of S to be from the same H-coset. The bound on |S| is improved for several classes of groups G, yielding optimal lower bounds for |S|. We also generalize Olson's result for |G|-term subsums to an analogous one for n-term subsums when n ≥ exp(G), with the bound likewise improved for several special classes of groups. This improves previous generalizations of Olson's result, with the bounds for n optimal.for any x ∈ A denotes the subgroup generated affinely by A, which is the smallest subgroup H such that A is contained in an H-coset. The relative complement of A is defined asWhen the subgroup H is implicit, it will usually be dropped from the notation. Regarding sequences and subsequence sums, we follow the standardized notation from Factorization Theory [4] [6] [11]. The key parts are summarized here. Let G 0 ⊆ G be a subset. A sequence S of terms from G 0 is viewed formally as an element of the free abelian monoid with basis G 0 , denoted F(G 0 ). Thus a sequence S ∈ F(G 0 ) is written as a finite multiplicative string of terms, using the bold dot operation · to concatenate terms, and with the order irrelevant: S = g 1 · . . . · g ℓ with g i ∈ G 0 the terms of S and |S| := ℓ ≥ 0 the length of S. Given g ∈ G 0 and s ≥ 0, we let g [s] = g · . . . · g s denote the sequence consisting of the element g repeated s times. We let v g (S) = |{i ∈ [1, ℓ] : g i = g}| ≥ 0 denote the multiplicity of the term g ∈ G 0 in the sequence S. If S, T ∈ F(G 0 ) are sequences, then S · T ∈ F(G 0 ) is the sequence obtained by concatenating the terms of T after those of S. A sequence S may also be defined by listing its terms as a product: S = • g∈G 0 g [vg(S)] . We use T | S to indicate that T is a subsequence of S and let T [−1] · S or S · T [−1] denote the sequence obtained by removing the terms of T from S. Then h(S) = max{v g (S) : g ∈ G 0 } is the maximum multiplicity of S, Supp(S) = {g ∈ G 0 : v g (S) > 0} ⊆ G is the support of S, σ(S) = ℓ i=1 g i = g∈G 0 v g (S)g ∈ G is the sum of S, Σ...

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Section: 2mentioning

confidence: 99%

Iterated sumsets and subsequence sums

Grynkiewicz

2018

Journal of Combinatorial Theory, Series A

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“…If A + B is aperiodic then one of the following holds: We shall also use the following extension of KST, recently obtained by Grynkiewicz [8], which describes the structure of pairs of sets (X, Y ) in an abelian group G verifying |X +Y | = |X|+|Y |. Again we only need a simplified version of the full result.…”

Section: ])mentioning

confidence: 99%

“…Theorem 6 (Grynkiewicz [8]). Let A and B be nonempty subsets of an abelian group G of odd order n verifying |A + B| = |A| + |B| ≤ |G| − 3.…”

Section: ])mentioning

confidence: 99%

Rainbow-free 3-colorings in abelian groups

Montejano

Serra

2009

Electronic Notes in Discrete Mathematics

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“…This notion is rediscovered by Balandraud in [1,2] where full subsets are named "cells", and also by Grynkiewicz [3] where the term "nonextendible subset" is used. The following lemma is straightforward: 2,7]).…”

Section: Introductionmentioning

confidence: 99%

On some subgroup chains related to Kneser’s theorem

Hamidoune¹,

Serra²,

Zémor³

2008

J. Théor. Nombres Bordeaux

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A recent result of Balandraud shows that for every subset S of an abelian group G there exists a non trivial subgroupThis strong form of Kneser's theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud's results in the abelian case.

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A Step Beyond Kemperman's Structure Theorem

Abstract: Abstract. We extend Kemperman's Structure Theorem by completely characterizing those finite subsets A and B of an arbitrary abelian group with |A + B| = |A| + |B|.

Cited by 20 publications

References 33 publications

Iterated sumsets and subsequence sums

Iterated sumsets and subsequence sums

Rainbow-free 3-colorings in abelian groups

On some subgroup chains related to Kneser’s theorem

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