Representation of finite abelian group elements by subsequence sums

Abstract

Let G ∼ = Cn 1 ⊕ · · · ⊕ Cn r be a finite and nontrivial abelian group with n 1 |n 2 |. .. |nr. A conjecture of Hamidoune says that if W = w 1 · · · wn is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S| ≥ |W | + |G| − 1 ≥ |G| + 1, the maximum multiplicity of S at most |W |, and σ(W) ≡ 0 mod |G|, then there exists a nontrivial subgroup H such that every element g ∈ H can be represented as a weighted subsequence sum of the form g = n P i=1 w i s i , with s 1 · · · sn a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W | ≥ 1 2 |G|. A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S| ≥ |G| + D(G) − 1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g + H such that all but at most |G/H| − 2 terms of S are from g + H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S| ≥ |G| + D(G) − 1 can be relaxed to |S| ≥ |G| + d * (G), where d * (G) = r P i=1 (n i − 1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least d * (G) of the w i are relatively prime to |G|. 1. Notation We follow the conventions of [9] for notation concerning sequences over an abelian group. For real numbers a, b ∈ R, we set [a, b] = {x ∈ Z | a ≤ x ≤ b}. Throughout, all abelian groups will be written additively. Let G be an abelian group, and let A, B ⊆ G be nonempty subsets. Then A + B = {a + b | a ∈ A, b ∈ B} denotes their sumset. The stabilizer of A is defined as H(A) = {g ∈ G | g + A = A}, and A is called periodic if H(A) = {0}, and aperiodic otherwise. If A is a union of H-cosets (i.e., H ≤ H(A)), then we say A is H-periodic. The order of an element g ∈ G is denoted ord(g), and we use φ H : G → G/H to denote the natural homomorphism. We use gcd(a, b) to denote the greatest common divisor of a, b ∈ Z. 2000 Mathematics Subject Classification. 11B75 (20K01).

A Step Beyond Kemperman's Structure Theorem

“…holds for d ≤ (M + ab)/ab. Thus, since M ≥ 0, then applying (14) with d = ⌊(M + ab)/ab⌋ ≥ 1 yields (iii). So it remains to establish (14).…”

“…Thus, since M ≥ 0, then applying (14) with d = ⌊(M + ab)/ab⌋ ≥ 1 yields (iii). So it remains to establish (14). Note that if l = 0, then |A + B| = |A||B| follows from (9), yielding (14).…”

“…So it remains to establish (14). Note that if l = 0, then |A + B| = |A||B| follows from (9), yielding (14). So we may assume l > 0.…”

“…Among other applications-including results in graph theory [20] and zero-sum additive theory [9]-Kemperman's Structure theorem (KST), whose statement we delay until later, yields the descriptions of those subsets of a locally compact abelian group whose Haar measure of the sumset fails to satisfy the triangle inequality [28,31]. Other applications may also be found in [13,14,33,36].…”

Normal sequences over finite abelian groups

“…To prove Theorem 1.5, we need the following result on the structure of the subgroup of a finite abelian group. [9,Proposition 4.4].) Let G be a finite abelian group, say, G = C n 1 ⊕ · · · ⊕ C n r with 1 < n 1 | · · · |n r .…”