Zero-sum problems in finite abelian groups and affine caps

For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form G = C_n^r, but they respect the structure of the group. In particular, we show s(C_n⁴) ≥ 20n - 19 for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.