On a Conjecture of Kemnitz

A classic theorem of Erdis, Ginzburg and Ziv states that in a sequence of 2n-1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let Z_nZ_n stand for the additive group of integers modulo n. Let s(n, d) denote the smallest integer s such that in any sequence of s elements from Z_n^dZ_n^d (the direct sum of d copies of Z_nZ_n) there is a subsequence of length n whose sum is 0 in Z_n^dZ_n^d. Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that s(p,2) ￡ 4p - 2s(p,2) \le 4p - 2 holds for every prime p. This implies that the value of s(p, 2) is either 4p-3 or 4p-2. For an arbitrary positive integer n it follows that s(n, 2) ￡ (41/10)ns(n, 2) \le (41/10)n. The proof uses an algebraic approach.