Inverse problems associated with subsequence sums in {{\displaystyle G}}_p\oplus {{\displaystyle C}}_p

Abstract

Let G be a finite abelian group and S be a sequence with elements of G: We say that S is a regular sequence over G if \left|{{\displaystyle S}}_H\right|\le \left|H\right|-\mathrm{1} holds for every proper subgroup H of G; where S_H denotes the subsequence of S consisting of all terms of S contained in H: We say that S is a zero-sum free sequence over G if \mathrm{0}\notin {\displaystyle \sum (S)}0; where {\displaystyle \sum (S)}\subset G denotes the set of group elements which can be expressed as a sum of a nonempty subsequence of S: In this paper, we study the inverse problems associated with {\displaystyle \sum (S)} when S is a regular sequence or a zero-sum free sequence over G={{\displaystyle G}}_p\oplus {{\displaystyle C}}_p, where p is a prime.

Keywords

Inverse problems / subsequence sums / regular sequences / zero-sum free sequences

References

[1] Bollobás B, Leader I. The number of k-sums modulo k: J Number Theory, 1999, 78: 27–35

[2] Eggletőn R B, Erdos P. Two combinatorial problems in group theory. Acta Arith, 1972, 21: 111–116

[3] Gao W, Hamidoune Y. On additive bases. Acta Arith, 1999, 88: 233–237

[4] Gao W, Han D, Qu Y, Qian G, Zhang H. On additive bases II. Acta Arith, 2015, 168: 247–267

[5] Gao W, Geroldinger A. On zero-sum sequences in ℤ/nℤ⊕ℤ/nℤ . Integers, 2003, 3: A08

[6] Gao W, Geroldinger A, Grynkiewicz D. Inverse zero-sum problems III. Acta Arith, 2010, 141: 103–152

[7] Gao W, Li Y,Peng J, Sun F.On subsequence sums of a zero-sum free sequence II. Electron J Combin, 2008, 15: R117

[8] Gao W, Qu Y, Zhang H. On additive bases III. Acta Arith, 2020, 193(3): 293–308

[9] Geroldinger A, Grynkiewicz D J. The large Davenport constant I: Groups with a cyclic, index 2 subgroup. J Pure Appl Algebra, 2013, 217: 863–885

[10] Geroldinger A, Halter-Koch F. Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory. Boca Raton: Chapman & Hall/CRC, 2006

[11] Guan H,Zeng X, Yuan P. Description of invariant F(5) of a zero-sum free sequence. Acta Sci Natur Univ Sunyatseni, 2010, 49: 1–4 (in Chinese)

[12] Martin G, Peilloux A, Wong E B. Lower bounds for sumsets of multisets in Zp2: Integers, 2013, 13: A72

[13] Matomäki K. On sumsets of multisets in Zpm : Electron J Combin, 2013, 20(3): P30

[14] Olson J E. A combinatorial problem on finite abelian groups I. J Number Theory, 1969, 1: 8–10

[15] Olson J E. A combinatorial problem on finite abelian groups II. J Number Theory, 1969, 2: 195–199

[16] Olson J E, White E T. Sums from a sequence of group elements . In: Zassenhaus H, ed. Number Theory and Algebra. New York: Academic Press, 1977, 215–222

[17] Peng C. Addition theorems in elementary abelian groups I. J Number Theory, 1987, 27: 46–57

[18] Peng J, Hui W. On the structure of zero-sum free set with minimum subset sums in abelian groups. Ars Combin, 2019, 146: 63–74

[19] Peng J, Li Y, Liu C,Huang M. On subsequence sums of a zero-sum free sequence over finite abelian groups. J Number Theory, 2020, 217: 193–217

[20] Peng J,Li Y,Liu C, Huang M. On the inverse problems associated with subsequence sums of zero-sum free sequences over finite abelian groups. Colloq Math (to appear)

[21] Pixton A. Sequences with small subsums sets. J Number Theory, 2009, 129: 806–817

[22] Qu Y, Han D. An inverse theorem for additive bases. Int J Number Theory, 2016, 12: 1509–1518

[23] Qu Y,Han D. Additive bases of Gp⊕Cpn. Int J Number Theory, 2017, 13: 2453–2459

[24] Qu Y, Wang G, Wang Q, Guo D. Extremal incomplete sets in finite abelian groups. Ars Combin, 2014, 116: 457–475

[25] Reiher C. A proof of the theorem according to which every prime number possesses Property B. Ph D Thesis. Rostock, 2010

[26] Sun F. On subsequence sums of a zero-sum free sequence. Electron J Combin, 2007, 14: R52

[27] Sun F,Peng J, Li Y.A note on the inverse problems associated with subsequence sums. J Combin Math Combin Comput (to appear)

[28] Yuan P. Subsequence sums of a zero-sumfree sequence. European J Combin, 2009, 30: 439–446

[29] Yuan P. Subsequence sums of zero-sum-free sequences. Electron J Combin, 2009, 16: R97 (13pp)

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Higher Education Press