Weierstrass gaps at n points of a curve

Fix a smooth curve C over a field K and n   distinct points P1,…,Pn∈C(K)P1,…,Pn∈C(K). Here we prove several results on the gap sequence of the n  -ple (P1,…,Pn)(P1,…,Pn) in two very different set-ups (either as a subset of NnNn or, as introduced by Beelen and Tutaş, as a subset of ZnZn). Most results are when K is algebraically closed and large.