On Pure k-sparse gapsets

TL;DR

This paper investigates the structure of gapsets with a focus on how the maximum distance between consecutive elements affects their properties, establishing a relationship between gapsets of different genera under certain conditions.

Contribution

It provides a new combinatorial relationship linking the cardinalities of gapsets with specified maximum distances across different genera, under the condition 2g ≤ 3κ.

Findings

Cardinality of gapsets with genus g and max distance κ equals that with genus g+1 and max distance κ+1 when 2g ≤ 3κ.

Establishes a recursive relationship between gapsets of different genera based on maximum distance.

Advances understanding of the combinatorial structure of gapsets in number theory.

Abstract

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 $2g \leq 3\kappa$.