Iterated sumsets and subsequence sums

Abstract

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...

Iterated sumsets and setpartitions

“…While the bounds given in Items 1-3 of Theorem 3.2 below are not tight, the worse-case scenario ones given in Items 1-4 above are, as can be seen by Examples B.1-B.3 in [35]. It seems to be a more challenging problem to find optimal bounds in Theorem 3.2 for n in terms of G/H, rather than G, particularly when G/H is not close to cyclic.…”

“…As was the case for Theorem 1.1, any of the following conditions combined with H < G being proper and nontrivial ensures that one of Items 1-3 holds in Theorem 3.2, and thus they can be substituted for Items 1-3 in Theorem 3. While the bounds given in Items 1-3 of Theorem 3.2 below are not tight, the worse-case scenario ones given in Items 1-4 above are, as can be seen by Examples B.1-B.3 in [35]. It seems to be a more challenging problem to find optimal bounds in Theorem 3.2 for n in terms of G/H, rather than G, particularly when G/H is not close to cyclic.…”

“…[35, Lemma 4.1] Let G be an abelian group, let n ≥ 1, let S ∈ F(G) be a sequence, let S ′ | S be a subsequence with h(S ′ ) ≤ n ≤ |S ′ |, let H = H(Σ n (S)), let X ⊆ G/H be the subset of all x ∈ G/H for which x has multiplicity at least n in φ H (S), and let Z = φ −1 H (X). Suppose |Σ n (S)| < |S ′ | − n + 1.…”

A Single Set Improvement to the $3k-4$ Theorem

“…Reducing modulo N and applying (8), it follows that i + j = i ′ + j ′ , in turn implying α i + β j = α i ′ + β j ′ per the definitions in (7).…”

“…The complete description is then addressed by the Kemperman Structure Theorem. We summarize the relevant details here, which may be found in [8,Chapter 9] and are summarized in more general form in [7] Let A, B ⊆ G and H ≤ G. A nonempty subset of the form (α + H) ∩ A is called an H-coset slice of A. If A ∅ ⊆ A is a nonempty subset of an H-coset and A \ A ∅ is H-periodic, then A ∅ is an H-coset slice and we say that A ∅ induces an H-quasi-periodic decomposition of A, namely, A = (A \ A ∅ ) ∪ A ∅ .…”