On Pure k-sparse gapsets

In this paper, we study gapsets and we focus on obtaining information on how the maximum distance between to consecutive elements influences the behaviour of the set. In particular, we prove that the cardinality of the set of gapsets with genus $g$ such that the maximum distance between two consecutive elements is $\kappa$ is equal to the cardinality of the set of gapsets with genus $g+1$ such that the maximum distance between two consecutive elements is $\kappa +1$, when $2g\le 3\kappa$.