On equitable zero sums

There is a rich literature on conditions guaranteeing that certain sums of residue classes cover certain other residue classes modulo some integer N . Often it is of particular importance for applications to know that the class 0 mod N can be represented as a sum of the studied residue classes, and the zero class is often the most difficult case. A well-known result along these lines is the famous Erdős-Ginzburg-Ziv theorem, which says that any sequence of 2N − 1 integers contains a subsequence of N integers whose sum is zero; furthermore, the example of N − 1 copies of 0 and N − 1 copies of 1 shows that all residue classes, except the zero class, can be represented if the sequence were only of length 2N − 2. In the study of sums of distinct residue classes modulo N (for example, the work of Olson [3]), the example a1 = 1, . . . , ar = r, with r = √ 4N − 1, shows that again the zero residue class is the most difficult to represent.