On the Erdös-Ginzburg-Ziv theorem

Let n be a natural number. Erdös, Ginzburg and Ziv proved that every sequence of elements of ℤn with length at least 2n - 1 contains an n-subsequence (subsequence of length n) with a zero sum. Generalizations of this result are obtained by Bialostocki-Dierker, Caro and Bialostocki-Lotspeich. We generalize the above result as follows. Let A be a set with cardinality at least 2n - 3 and f : A → ℤn; then either • there exists an n-subset S ⊆ A such that ∑xεSf(x) = 0 or • there are a, b εℤn such that ℤn is generated by b - a and | f−1 (a)| = n - 1 and one of the following conditions hold. (i) |A| ⩽2n − 2 and | f−1 (b)| = |A| − n + 1. (ii) |A| ⩽2 n − 2 and | f−1 (b)| = n − 3 and |f−1 (2b − a)| =1. Let a, u εℤn be such that ℤn is generated by μ. Clearly, the sequence contains exactly one n-subsequence with zero sum. We show that every (2n - 1)-sequence in ℤn which is not of this type contains 5 n-subsequences with a zero sum.