Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Grynkiewicz, David J.

doi:10.1007/978-3-030-67996-5_11

Cited by 2 publications

(2 citation statements)

References 36 publications

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“…The goal of this section is to prove Theorem 4.1. To do so, we make use of the following recent strengthenings of the Partition Theorem, formulated as Propositions 4.2 and 4.3, as well as a basic lemma from [23]. Their proofs are given in [23, Theorem 3.2, Theorem 3.3, Lemma 2.4] (for simplicity, we state here only the cyclic case in Proposition 4.…”

Section: Algebraic Propertiesmentioning

confidence: 99%

“…We do not involve algebraic considerations (structural results on the class semigroup of monoids of product-one sequences were recently established in [31,Section 3]) but work with methods from additive combinatorics. We use substantially the recent characterization of minimal product-one sequences of maximal length (Proposition 2.4) and a recent refinement ( [23]) of the Partition Theorem ([26, Chapters 14 and 15]). Let G be a dihedral group of order 2n for some odd n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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On product-one sequences over dihedral groups

Geroldinger

Grynkiewicz

et al. 2019

Preprint

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Let G be a finite group. A sequence over G means a finite sequence of terms from G, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over G (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

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Section: Algebraic Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On product-one sequences over dihedral groups

Geroldinger

Grynkiewicz

et al. 2019

Preprint

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A Multiplicative Property for Zero-Sums II

Grynkiewicz¹,

Liu²

2022

Electron. J. Combin.

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For $n\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\oplus C_{mn}$ with $n\geq 2$ and $m\geq 1$, and let $k\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to characterize those sequences with maximal length $mn+n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $mn+n-1-k$. For $k\leq 1$, this is the inverse question for the Davenport Constant. For $k=n-1$, this is the inverse question for the $\eta(G)$ invariant concerning short zero-sum subsequences. For $C_n\oplus C_n$ and $k\in [2,n-2]$, with $n\geq 5$ prime, it was conjectured in a paper of Grynkiewicz, Wang and Zhao that they must have the form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[n-1]}\boldsymbol{\cdot} (e_1+e_2)^{[k]}$ for some basis $(e_1,e_2)$, with the conjecture established in many cases and later extended to composite moduli $n$. In this paper, we focus on the case $m\geq 2$. Assuming the conjectured structure holds for $k\in [2,n-2]$ in $C_n\oplus C_n$, we characterize the structure of all sequences of maximal length $mn+n-2+k$ in $C_n\oplus C_{mn}$ that fail to contain a nontrivial zero-sum of length at most $mn+n-1-k$, showing they must either have the form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[sn-1]}\boldsymbol{\cdot} (e_1+e_2)^{[(m-s)n+k]}$ for some $s\in [1,m]$ and basis $(e_1,e_2)$ with $\mathsf{ord}(e_2)=mn$, or else have the form $S=g_1^{[n-1]}\boldsymbol{\cdot} g_2^{[n-1]}\boldsymbol{\cdot} (g_1+g_2)^{[(m-1)n+k]}$ for some generating set $\{g_1,g_2\}$ with $\mathsf{ord}(g_1+g_2)=mn$. In view of prior work, this reduces the structural characterization for a general rank two abelian group to the case $C_p\oplus C_p$ with $p$ prime. Additionally, we give a new proof of the precise structure in the case $k=n-1$ for $m=1$. Combined with known results, our results unconditionally establish the structure of extremal sequences in $G=C_n\oplus C_{mn}$ in many cases, including when $n$ is only divisible by primes at most $7$, when $n\geq 2$ is a prime power and $k\leq \frac{2n+1}{3}$, or when $n$ is composite and $k=n-d-1$ or $n-2d+1$ for a proper, nontrivial divisor $d\mid n$.

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Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Cited by 2 publications

References 36 publications

On product-one sequences over dihedral groups

On product-one sequences over dihedral groups

A Multiplicative Property for Zero-Sums II

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