Rocky Mountain Journal of Mathematics

On the existence of zero-sum subsequences of distinct lengths

Benjamin Girard

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Rocky Mountain J. Math., Volume 42, Number 2 (2012), 583-596.

First available in Project Euclid: 23 April 2012

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Primary: 11R27: Units and factorization 11B75: Other combinatorial number theory 11P99: None of the above, but in this section 20D60: Arithmetic and combinatorial problems 20K01: Finite abelian groups [For sumsets, see 11B13 and 11P70] 05E99: None of the above, but in this section 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations


Girard, Benjamin. On the existence of zero-sum subsequences of distinct lengths. Rocky Mountain J. Math. 42 (2012), no. 2, 583--596. doi:10.1216/RMJ-2012-42-2-583.

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  • N. Alon, Combinatorial Nullstellensatz, Comb. Probab. Comput. 8 (1999), 7-29.
  • N. Alon, S. Friedland and G. Kalai, Regular subgraphs of almost regular graphs, J. Combin. Theory 37 (1984), 79-91.
  • P. van Emde Boas and D. Kruyswijk, A combinatorial problem on finite abelian groups, Reports ZW-1967-009, Math. Centre, Amsterdam, 1967.
  • P. Erdős and E. Szemerédi, On a problem of Graham, Publ. Math. Debrecen 23 (1976), 123-127.
  • W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996), 100-103.
  • –––, On zero-sum subsequences of restricted size II, Discrete Math. 271 (2003), 51-59.
  • W.D. Gao and J.J. Zhuang, Sequences not containing long zero-sum subsequences, Europ. J. Comb. 27 (2006), 777-787.
  • W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey, Expo. Math. 24 (2006), 337-369.
  • W. Gao, A. Geroldinger and D. Grynkiewicz, Inverse zero-sum problems III, Acta Arith. 141 (2010), 103-152.
  • W. Gao, A. Geroldinger and W.A. Schmid, Inverse zero-sum problems, Acta Arith. 128 (2007), 245-279.
  • W. Gao, Y. Hamidoune and G. Wang, Distinct lengths modular zero-sum subsequences: A proof of Graham's conjecture, J. Number Theory 130 (2010), 1425-1431.
  • A. Geroldinger, Additive group theory and non-unique factorizations, in Combinatorial number theory and additive group theory, A. Geroldinger and I. Ruzsa, eds., Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser, 2009.
  • A. Geroldinger and F. Halter-Koch, Non-unique factorizations. Algebraic, combinatorial and analytic theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton, 2006.
  • D. Grynkiewicz, Note on a conjecture of Graham, Europ. J. Comb. 32 (2011), 1336-1344.
  • H. Guan, P. Yuan and X. Zeng, Normal sequences over finite abelian groups, J. Combin. Theory 118 (2011), 1519-1524.
  • J.E. Olson, A combinatorial problem on finite abelian groups, I, J. Number Theory 1 (1969), 8-10.
  • C. Reiher, A proof of the theorem according to which every prime number possesses Property B, manuscript.
  • S. Savchev and F. Chen, Long $n$-zero-free sequences in finite cyclic groups, Discrete Math. 308 (2008), 1-8.
  • W.A. Schmid, On zero-sum subsequences in finite abelian groups, Integers 1 (2001), #A01.
  • –––, Inverse zero-sum problems II, Acta Arith. 143 (2010), 333-343.