## Representation of finite abelian group elements by subsequence sums.(English)Zbl 1214.11034

Let $$G$$ be a finite abelian group. Hamidoune conjectures that if $$W=w_1\cdot\dots\cdot w_n$$ is a sequence of integers, all but at most one relatively prime to $$|G|$$, and $$S$$ is a sequence over $$G$$ with $$|S|\geq |W|+|G|-1\geq |G|+1$$, the maximum multiplicity of $$S$$ at most $$|W|$$, and $$\sigma(W)\equiv 0 \pmod {|G|}$$, then there exists a nontrivial subgroup $$H$$ such that every element $$g\in H$$ can be represented as a weighted subsequence sum of the form $$g=\sum_{i=1}^nw_is_i$$, with $$s_1\cdot\dots\cdot s_n$$ a subsequence of $$S$$. The authors give two examples showing this does not hold in general, and characterize the counterexamples for large $$|W|\geq \frac{1}{2}|G|$$.
A theorem of Gao states that if $$S$$ is a sequence over $$G$$ with $$|S|\geq |G|+D(G)-1$$, 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 $$g+H$$. The authors 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 they use to also improve the previously mentioned result of Gao by showing that the hypothesis $$|S|\geq |G|+D(G)-1$$can be relaxed to $$|S|\geq |G|+d^*(G)$$, where $$d^*(G)=\sum_{i=1}^r (n_i-1)$$. They also use this method to derive a variation of Hamidoune’s conjecture valid when at least $$d^*(G)$$ of the $$w_i$$ are relatively prime to $$|G|$$.

### MSC:

 11B75 Other combinatorial number theory 20K01 Finite abelian groups
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### References:

 [1] Sukumar das Adhikari and Purusottam Rath, Davenport constant with weights and some related questions. Integers 6 (2006), A30, 6 pp (electronic). · Zbl 1107.11018 [2] Sukumar das Adhikari and Yong-Gao Chen, Davenport constant with weights and some related questions II. J. Combin. Theory Ser. A 115 (2008), no. 1, 178-184. · Zbl 1210.11031 [3] N. Alon, A. Bialostocki and Y. Caro, The extremal cases in the Erdős-Ginzburg-Ziv Theorem. Unpublished. [4] A. Bialostocki, P. Dierker, D. J. Grynkiewicz, and M. Lotspeich, On Some Developments of the Erdős-Ginzburg-Ziv Theorem II. Acta Arith. 110 (2003), no. 2, 173-184. · Zbl 1069.11007 [5] Y. Caro, Zero-sum problems—a survey. Discrete Math. 152 (1996), no. 1-3, 93-113. · Zbl 0856.05068 [6] P. Erdős, A. Ginzburg and A. Ziv, Theorem in Additive Number Theory. Bull. Res. Council Israel 10F (1961), 41-43. · Zbl 0063.00009 [7] W. Gao, Addition theorems for finite abelian groups. J. Number Theory 53 (1995), 241-246. · Zbl 0836.11007 [8] W. Gao and A. Geroldinger, On Long Minimal Zero Sequences in Finite Abelian Groups. Periodica Math. Hungar. 38 (1999), no. 3, 179-211. · Zbl 0980.11014 [9] W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey. Expositiones Mathematicae, 24 (2006), no. 4, 337-369. · Zbl 1122.11013 [10] W. Gao and W. Jin, Weighted sums in finite cyclic groups. Discrete Math. 283 (2004), no. 1-3, 243-247. · Zbl 1052.11014 [11] A. Geroldinger and F. Halter-Koch, Non-unique factorizations: Algebraic, combinatorial and analytic theory. Pure and Applied Mathematics (Boca Raton) 278. Chapman & Hall/CRC, Boca Raton, FL, 2006. · Zbl 1113.11002 [12] A. Geroldinger and R. Schneider, On Davenport’s Constant. J. Combin. Theory, Ser. A 61 (1992), no. 1, 147-152. · Zbl 0759.20008 [13] S. Griffiths, The Erdős-Ginzberg-Ziv theorem with units. To appear in Discrete math. · Zbl 1206.11032 [14] D. J. Grynkiewicz, A Weighted Erdős-Ginzburg-Ziv Theorem. Combinatorica 26 (2006), no. 4, 445-453. · Zbl 1121.11018 [15] D. J. Grynkiewicz, Quasi-periodic Decompositions and the Kemperman Structure Theorem, European J. Combin. 26 (2005), no. 5, 559-575. · Zbl 1116.11081 [16] D. J. Grynkiewicz, On a Partition Analog of the Cauchy-Davenport Theorem. Acta Math. Hungar. 107 (2005), no. 1-2, 161-174. · Zbl 1102.11016 [17] D. J. Grynkiewicz, On a conjecture of Hamidoune for subsequence sum., Integers 5 (2005), no. 2, A7, 11 pp. (electronic). · Zbl 1098.11019 [18] D. J. Grynkiewicz and R. Sabar, Monochromatic and zero-sum sets of nondecreasing modified diameter. Electron. J. Combin. 13 (2006), no. 1, Research Paper 28, 19 pp. (electronic). · Zbl 1084.05073 [19] D. J. Grynkiewicz, Sumsets, Zero-sums and Extremal Combinatorics. Ph. D. Dissertation, Caltech (2005). [20] D. J. Grynkiewicz, A Step Beyond Kemperman’s Stucture Theorem. Preprint (2007). [21] Y. O. Hamidoune and A. Plagne, A new critical pair theorem applied to sum-free sets in abelian groups. Comment. Math. Helv. 79 (2004), no. 1, 183-207. · Zbl 1045.11072 [22] Y. O. Hamidoune, On weighted sequence sums. Comb. Prob. Comput. 4 (1995), 363-367. · Zbl 0848.20049 [23] Y. O. Hamidoune, On weighted sums in abelian groups. Discrete Math. 162 (1996), 127-132. · Zbl 0872.11016 [24] T. Hungerford, Algebra. Springer-Verlag, New York, 1974. · Zbl 0293.12001 [25] J. H. B. Kemperman, On Small Sumsets in an Abelian Group. Acta Math. 103 (1960), 63-88. · Zbl 0108.25704 [26] M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58 (1953), 459-484. · Zbl 0051.28104 [27] M. Kneser, Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen. Math. Z. 64 (1955), 429-434. · Zbl 0064.04305 [28] S. Lang, Algebra. Third edition, Yale University, New Haven, CT, 1993. [29] V. Lev, Critical pairs in abelian groups and Kemperman’s structure theorem. Int. J. Number Theory 2 (2006), no. 3, 379-396. · Zbl 1157.11040 [30] M. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics 165, Springer-Verlag, New York, 1996. · Zbl 0859.11003 [31] J. E. Olson, An addition theorem for finite abelian groups. J. Number Theory 9 (1977), no. 1, 63-70. · Zbl 0351.20032 [32] O. Ordaz and D. Quiroz, Representation of group elements as subsequences sums. Discrete Mathematics 308 (2008), no. 15, 3315-3321. · Zbl 1143.20032 [33] T. Tao and V. Vu, Additive Combinatorics. Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, Cambridge, 2006. · Zbl 1127.11002
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