Davenport constant of a box in \(\mathbb{Z}^2\). (English) Zbl 1459.11069
Summary: Let \(X\) be a subset of an abelian group \(G\). Then a sequence \(S\) over \(X\) is called a zero-sum sequence if the sum of \(S\) is zero, and a minimal zero-sum sequence if it is a non-empty zero-sum sequence such that all proper subsequences are not zero-sum sequences. The Davenport constant of \(X\), denoted by \(\mathsf{D}(X)\), is defined as the supremum of lengths of minimal zero-sum sequences over \(X\). In this paper, we investigate minimal zero-sum sequences over \([[ -1,m]]\times[[-1,n]]\subset\mathbb{Z}^2\). We completely determine the structure of minimal zero-sum sequences of length at least \((m+1)(n+1)\), and hence derive that \(\mathsf{D}([[-1,m]]\times [[-1,n]])=(m+1)(n+1)\).
MSC:
| 11B75 | Other combinatorial number theory |
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
| 11P70 | Inverse problems of additive number theory, including sumsets |
References:
| [1] | D. F. Anderson, S. T. Chapman and W. W. Smith,Some factorization properties of Krull domains with infinite cyclic divisor class group, J. Pure Appl. Algebra 96 (1994), 97-112. · Zbl 0831.13001 |
| [2] | N. R. Baeth and A. Geroldinger,Monoids of modules and arithmetic of direct-sum decompositions, Pacific J. Math. 271 (2014), 257-319. · Zbl 1347.16006 |
| [3] | N. R. Baeth, A. Geroldinger, D. J. Grynkiewicz and D. Smertnig,A semigrouptheoretical view of direct-sum decompositions and associated combinatorial problems, J. Algebra Appl. 14 (2015), art. 1550016, 60 pp. · Zbl 1385.16003 |
| [4] | P. Baginski, S. T. Chapman, R. Rodriguez, G. J. Schaeffer and Y. She,On the Delta set and catenary degree of Krull monoids with infinite cyclic divisor class group, J. Pure Appl. Algebra 214 (2010), 1334-1339. · Zbl 1193.20071 |
| [5] | G. Deng and X. Zeng,Long minimal zero-sum sequences over a finite subset ofZ, Eur. J. Combin. 67 (2018), 78-86. · Zbl 1371.05303 |
| [6] | W. Gao and A. Geroldinger,Zero-sum problems in abelian groups: a survey, Expo. Math. 24 (2006), 337-369. · Zbl 1122.11013 |
| [7] | A. Geroldinger, D. J. Grynkiewicz, G. J. Schaeffer and W. A. Schmid,On the arithmetic of Krull monoids with infinite cyclic class group, J. Pure Appl. Algebra 214 (2010), 2219-2250. · Zbl 1208.13003 |
| [8] | A. Geroldinger and F. Halter-Koch,Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, 2006. · Zbl 1113.11002 |
| [9] | A. Geroldinger and I. Z. Ruzsa,Combinatorial Number Theory and Additive Group Theory, Adv. Courses Math. CRM Barcelona, Birkhäuser, 2009. · Zbl 1177.11005 |
| [10] | B. Girard,An asymptotically tight bound for the Davenport constant, J. École Polytech. Math. 5 (2018), 605-611. · Zbl 1401.05311 |
| [11] | D. J. Grynkiewicz,Structural Additive Theory, Dev. Math. 30, Springer, 2013. · Zbl 1368.11109 |
| [12] | W. Hassler,Factorization properties of Krull monoids with infinite class group, Colloq. Math. 92 (2002), 229-242. · Zbl 0997.20056 |
| [13] | C. Liu,On the lower bounds of Davenport constant, J. Combin. Theory Ser. A 171 (2020), art. 105162, 15 pp. · Zbl 1446.11042 |
| [14] | W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, 2004. · Zbl 1159.11039 |
| [15] | A. Plagne and S. Tringali,The Davenport constant of a box, Acta Arith. 171 (2015), 197-219. · Zbl 1384.11042 |
| [16] | P. A. Sissokho,A note on minimal zero-sum sequences overZ, Acta Arith. 166 (2014), 279-288. · Zbl 1369.11021 |
| [17] | X. Zeng and G. Deng,Minimal zero-sum sequences overJ−m, nK, J. Number Theory 203 (2019), 230-241. · Zbl 1446.11044 |
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