Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group. II. (English) Zbl 1503.11018
Let \((G,+)\) be a finite abelian group with identity element \(0\). Denote by \(\mathcal{F}(G)\) the set of finite sequences over \(G\). For a sequence \(T\in\mathcal{F}(G)\) and an element \(g\in G\) denote by \(v_g(T)\) the multiplicity of \(g\) in \(T\).
From 1960 onwards, works on the existence of sequences or subsequences with sum zero were done and some related parameters (or “constants”) were defined: the Davenport constant, the Erdős-Ginzburg-Ziv constant, or parameters suggested by Olson and more recently by Girard.
W. Gao et al. [Int. J. Number Theory 14, No. 3, 705–711 (2018; Zbl 1415.11045)] invented a different approach to this type of problems by introducing new concepts whose special cases are the classical parameters.
In the two papers under review (Part I, Part II), the authors go further to the study of invariants introduced in [loc. cit.]. Here is a short overview of the main results.
Part I. – Denote by \(\mathcal{B}(G)\) the subset of \(\mathcal{F}(G)\) formed by the zero-sum sequences \(T\in\mathcal{F}(G)\). For \(\Omega\subseteq\mathcal{B}(G)\), let \(d_{\Omega}(G)\) be the smallest integer \(t\) such that every sequence \(S\in\mathcal{F}(G)\) of length \(|S|\geq t\) has a subsequence in \(\Omega\). Let \(q'(G)\) be the smallest integer \(t\) such that every sequence \(S\in\mathcal{F}(G)\) of length \(|S|\geq t\) has two nonempty zero-sum subsequences \(T_1\) and \(T_2\) having different forms, that is \(v_g(T_1)\neq v_g(T_2)\) for at least one \(g\in G\). Let \(q(G)\) be the smallest integer \(t\) such that \[ \bigcap_{d_\Omega(G)=t}\Omega =\emptyset. \] The invariants \(d_{\Omega}(G)\), \(q'(G)\) and \(q(G)\) were introduced in [loc. cit.]. Here further properties of these invariants are proved. One of them is that \(q'(G)=q(G)\).
Part II. – A sequence \(S\in\mathcal{F}(G)\) is called week-regular if, for every \(g\in G\), \(v_g(S)\leq\mathrm{ord}(g)\). Let \(N(G)\) denote the smallest integer \(t\) such that every weak-regular sequence \(S\in\mathcal{F}(G)\) of length \(|S|\geq t\) has a nontrivial zero-sum subsequence \(T\) satisfying \(v_g(T)=v_g(S)>0\) for at least one \(g\) appearing in \(S\). In Part II, among other results, the authors determine the value of \(N(G)\) for cyclic groups of prime power order and for elementary abelian groups of rank two.
From 1960 onwards, works on the existence of sequences or subsequences with sum zero were done and some related parameters (or “constants”) were defined: the Davenport constant, the Erdős-Ginzburg-Ziv constant, or parameters suggested by Olson and more recently by Girard.
W. Gao et al. [Int. J. Number Theory 14, No. 3, 705–711 (2018; Zbl 1415.11045)] invented a different approach to this type of problems by introducing new concepts whose special cases are the classical parameters.
In the two papers under review (Part I, Part II), the authors go further to the study of invariants introduced in [loc. cit.]. Here is a short overview of the main results.
Part I. – Denote by \(\mathcal{B}(G)\) the subset of \(\mathcal{F}(G)\) formed by the zero-sum sequences \(T\in\mathcal{F}(G)\). For \(\Omega\subseteq\mathcal{B}(G)\), let \(d_{\Omega}(G)\) be the smallest integer \(t\) such that every sequence \(S\in\mathcal{F}(G)\) of length \(|S|\geq t\) has a subsequence in \(\Omega\). Let \(q'(G)\) be the smallest integer \(t\) such that every sequence \(S\in\mathcal{F}(G)\) of length \(|S|\geq t\) has two nonempty zero-sum subsequences \(T_1\) and \(T_2\) having different forms, that is \(v_g(T_1)\neq v_g(T_2)\) for at least one \(g\in G\). Let \(q(G)\) be the smallest integer \(t\) such that \[ \bigcap_{d_\Omega(G)=t}\Omega =\emptyset. \] The invariants \(d_{\Omega}(G)\), \(q'(G)\) and \(q(G)\) were introduced in [loc. cit.]. Here further properties of these invariants are proved. One of them is that \(q'(G)=q(G)\).
Part II. – A sequence \(S\in\mathcal{F}(G)\) is called week-regular if, for every \(g\in G\), \(v_g(S)\leq\mathrm{ord}(g)\). Let \(N(G)\) denote the smallest integer \(t\) such that every weak-regular sequence \(S\in\mathcal{F}(G)\) of length \(|S|\geq t\) has a nontrivial zero-sum subsequence \(T\) satisfying \(v_g(T)=v_g(S)>0\) for at least one \(g\) appearing in \(S\). In Part II, among other results, the authors determine the value of \(N(G)\) for cyclic groups of prime power order and for elementary abelian groups of rank two.
Reviewer: Georges Grekos (St. Étienne)
MSC:
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 11R27 | Units and factorization |
| 11P70 | Inverse problems of additive number theory, including sumsets |
| 20K01 | Finite abelian groups |
References:
| [1] | Erdős, P., Über die Primzahlen gewisser arithmetischer Reihen, Math. Z., 39, 1, 473-491 (1935), (in German) · JFM 61.0135.03 |
| [2] | Gao, W.; Geroldinger, A., Zero-sum problems in finite abelian groups: a survey, Expo. Math., 24, 337-369 (2006) · Zbl 1122.11013 |
| [3] | Gao, W.; Han, D.; Qian, G.; Qu, Y.; Zhang, H., On additive bases II, Acta Arith., 168, 3, 247-267 (2015) · Zbl 1330.11064 |
| [4] | Gao, W.; Hong, S.; Hui, W.; Li, X.; Yin, Q.; Zhao, P., Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group, Period. Math. Hung. (2021) |
| [5] | Gao, W.; Huang, M.; Hui, W.; Li, Y.; Liu, C.; Peng, J., Sums of sets of abelian group elements, J. Number Theory, 208, 208-229 (2020) · Zbl 1464.11031 |
| [6] | Gao, W.; Li, Y.; Peng, J.; Wang, G., A unifying look at zero-sum invariants, Int. J. Number Theory, 14, 705-711 (2018) · Zbl 1415.11045 |
| [7] | Geroldinger, A., Additive group theory and non-unique factorizations, (Geroldinger, A.; Ruzsa, I., Combinatorial Number Theory and Additive Group Theory. Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics, CRM Barcelona (2009), Birkhäuser), 1-86 · Zbl 1221.20045 |
| [8] | Geroldinger, A.; Halter-Koch, F., Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math., vol. 278 (2006), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton · Zbl 1113.11002 |
| [9] | Geroldinger, A.; Halter-Koch, F.; Zhong, Q., On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms · Zbl 1524.20058 |
| [10] | Hamidoune, Y. O.; Zémor, G., On zero-free subset sums, Acta Arith., 78, 143-152 (1996) · Zbl 0863.11016 |
| [11] | Nathanson, M. B., Additive Number Theory: Inverse Problems and the Geometry of Sumsets, GTM, vol. 165 (1996), Springer · Zbl 0859.11003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
