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Journal of Combinatorial Theory, Series A 95, 387389 (2001) doi:10.1006jcta.2001.3181, available online at http:www.idealibrary.com on

NOTE Note on a Zero-Sum Problem W. Gao 1 Department of Computer Science and Technology, University of Petroleum, Beijing 102200, China Communicated by the Managing Editors Received November 1, 2000; published online June 18, 2001

Let G be an additively written, finite abelian group, and exp(G ) its exponent. Let S=(a 1 , ..., a k ) be a sequence of elements in G; we say that S is a zero-sum sequence if  ki=1 a i =0. Let s(G ) be the samllest integer t such that every sequence of t elements in G contains a zero-sum subsequence of length exp(G ). This constant has been studied by serveral authors during last 20 years [19, 11]. Let C n be the cyclic group of order n, and C kn the direct product of k copies of C n . In [3], Erdo s et al. proved that S(C 2n )=2n&1. A geometrical interpretation of s(C dn ) was given by Harborth in [8]. In 1980, Kemntiz suggested the following Conjecture 1. s(C 2n )=4n&3. Alon and Dubiner [1] showed that s(C 2n )6n&5. The author [5] showed that s(C 2n )=4n&3 for n=2 a3 b5 c7 dm, where a, b, c, d are nonnegative integers and m(2 a+23 b&15 c7 d ) 13. Recently, Ronyai [11] proved that s(C 2p )4p&2 for every prime p. In this paper we obtain the following Theorem 2.

If p is a prime then s(C 2p k )4p k &2.

Let G be a finite abelian group. By * we denote the empty sequence and adopt the convention that * is a zero-sum sequence. T/S means that T is a subsequence of S. By f E (S) ( f O(S)) we denote the number of zero-sum subsequences T of S with 2 | |T | (2 |% |T | ). Clearly, f E (S) f E (*)=1. Lemma 3 [10]. Let p be a prime, and S a sequence of elements in C np k . Suppose that |S| n( p k &1)+1. Then, f E (S)# f O(S) (mod p). 1 This work has been supported partly by National Natural Science Foundation of China and the Foundation of University of Petroleum.

387 0097-316501 35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

388

NOTE

Lemma 4 [1]. If S is a zero-sum sequence of 3p k elements in C 2p k then S contains a zero-sum subsequence of length p k. Alon and Dubiner proved Lemma 4 for k=1 but their method works for all k. Let T be a sequence of elements in C 2p k . By r(T ) we denote the number of zero-sum subsequences W of T with |W | =2p k. Lemma 5. Let T be a sequence of elements in C 2p k with 3p k &2 |T |  4p k &1. Suppose that T contains no zero-sum subsequence of length p k. Then, r(T )#&1 (mod p). Proof. Set t= |T |. Suppose T=(b 1 , ..., b t ). Set c i =(1, b i ) with 1 # C p k for i=1, ..., t. Then, c i # C 3p k . Put U=(c 1 , ..., c t ). Let V be a zero-sum subsequence of U. We clearly have, p k | |V |. Since, |V| = |T| 4p k &1, |V | = p k, 2p k or 3p k. It follows from Lemma 4 that |V | =2p k. Now this lemma follows from Lemma 3. K Proof of Theorem 2. Assume to the contrary that, S contains no zerosum subsequence of length p k. By Lemma 5 we conclude that r(T)# &1

(mod p)

holds for every subsequence T of S with |T | 3p k &2. We clearly have r(T )=

: T/S, |T | =3p k &2

\

4p k &2&2p k r(S). 3p k &2&2p k

+

Therefore, (&1)#

: T/S, |T | =3p k &2

\

2p k &2 (&1) p k &2

+

(mod p).

This gives that

\

4p k &2 2p k &2 # 3p k &2 p k &2

+ \

+

(mod p).

Therefore, 3#

\

4p k &2 4p k &2 2p k &2 2p k &2 # # # #1 pk 3p k &2 p k &2 pk

+ \

+ \

+ \

a contradiction. This completes the proof.

+

(mod p),

NOTE

389

Remark 6. The problem to determine s(C dn ) for d3 and n>2 remains widely open. Alon and Dubiner obtained that s(C dn )(cd log d ) d n [2]; the author and Yang [7] showed that s(C dn )n d +n&1. It was conjectured [2] that s(C dn )c dn for some absolute constant c.

ACKNOWLEDGMENTS I thank the referee and Professor A. Geroldinger for their useful suggestions and comments.

REFERENCES 1. N. Alon and M. Dubiner, Zero-sum sets of prescribed size, in ``Combinatorics, Paul Erdo s Is Eighty. Vol. 1. Keszthely,'' pp. 3350, Bolyai Soc. Math. Stud., Janos Bolyai Math. Soc., Budapest, 1993. 2. N. Alon and M. Dubiner, A lattice point problem and additive number theory, Combinatorica 15 (1995), 301309. 3. P. Erdo s, A. Ginzburg, and A. Ziv, A theorem in the additive number theory, Bull. Res. Council. Israel 10 (1961), 4143. 4. W. D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996), 100103. 5. Weidong Gao, On zero-sum subsequences of restricted size, J. Number Theory 61 (1996), 97102. 6. W. Gao, Two zero-sum problems and multiple properties, J. Number Theory 81 (2000), 254265. 7. W. Gao and Y. Yang, Note on a combinatorial constant, J. Math. Res. Exposition 17 (1997), 139140. 8. H. Harborth, Ein Extremaproblem fur Gitterpunkte, J. Reine Angew. Math. 262 263 (1973), 356360. 9. A. Kemnitz, On a lattice point problem, Ars Combin. 16 (1983), 151160. 10. J. E. Olson, A combinatorial problem on finite abelian groups, I, J. Number Theory 1 (1969), 811. 11. L. Ronyai, On a conjecture of Kemnitz, Combinatorica 20 (2000), 569573.