A Short Survey on Upper and Lower Bounds for Multidimensional Zero Sums

After giving some background on sums of residue classes we explained the following problem on multidimensional zero sums which is well known in combinatorial number theory: Let f(n, d) denote the least integer such that any choice of f(n, d) elements in Zn contains a subset of size n whose sum is zero. Harborth [12] proved that (n − 1)2 + 1 ≤ f(n, d) ≤ (n − 1)n + 1. The lower bound follows from the example in which there are n − 1 copies of each of the 2 vectors with entries 0 or 1. The upper bound follows since any set of (n − 1)n + 1 vectors must contain, by the pigeonhole principle, n vectors which are equivalent modulo n. If d is fixed, the upper bound was improved considerably by Alon and Dubiner [2] to cd n. Erdős, Ginzburg, and Ziv [6] proved that f(n, 1) = 2n−1 and Kemnitz conjectured that f(n, 2) = 4n− 3. There are partial results due to Kemnitz [14], as well as Gao [8], [9], [10], [11], Rónyai [16] and Thangadurai [17]. For example, Rónyai [16] proved that for primes p one has f(p, 2) ≤ 4p− 2, which implies that f(n, 2) ≤ 4.1n. Gao [11] extended this to powers of primes: f(p, 2) ≤ 4p − 2. If n is fixed but d is increasing not very much is known.