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On the number of zero sum subsequences

✍ Scribed by Weidong Gao

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
219 KB
Volume
163
Category
Article
ISSN
0012-365X

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✦ Synopsis

Let Z. be the cyclic group of order n. For a sequence S of elements in Z~, we use f~(S) to denote the number of subsequences, the sum of whose elements is zero. In this paper, we give a characterization on the sequences S of elements in Zn for whichf~(S) < 2 Isl -" Γ· k -,, under the restriction 1 ~ k ~< Fn/4] + i. As consequences of this result, we obtain some further characterizations on the sequences S of n elements in Z. for which f.(S) is not large. Let Z. be the cyclic group of order n. For a sequence S = (at ..... ak) of elements in k Zn, we use ~ S to denote the sum ~ ~= t a~. By 2 we denote the empty sequence and adopt the convention that Y. 2 --0. We usef~(S) to denote the number of subsequences T of S with ~ T = 0. Clearly, f~(S) ~>f~(2) = 1. In ['3], to give a positive answer to a conjecture of Erd6s, Olson proved that if al ..... a, is a sequence of n nonzero elements in Zn, and if al ..... an are not all equal, thenfn(S) >I n. In this paper, we first give a characterization on the sequences S of elements in Zn with f~(S) < 2 Isl -" Γ· k-1 under the restriction 1 ~< k ~< ['n/4] + 1. As consequences of this result, we get some further characterizations on the sequences S of n elements in Z. for whichf.(S) is not large. Among these characterizations, we generalize both the result of Oson and a result of Bulman-Fleming and Wang.

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