Lower bounds for multidimensional zero sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in ℤ_n ^d contains a subset of size n whose sum is zero. Harborth proved that (n - 1)2^d + 1 ≤ f(n, d) ≤ (n - 1)n^d + 1. The upper bound was improved by Alon and Dubiner to c _dn. It is known that f(n, 1) = 2n - 1 and Reiher proved that f(n, 2) = 4n - 3. Only for n = 3 it was known that f(n, d) > (n - 1)2^d + 1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p, d). In this note we show that this is not the case. In fact, for all odd n ≥ 3 and d ≥ 3 we show that f(n, d) ≥ 1.125^⌊d/3⌋(n - 1)2^d + 1.