Equidistribution of numerical semigroup gaps modulo $m$

TL;DR

This paper investigates when the gaps of a numerical semigroup are evenly distributed across residue classes modulo m, especially focusing on cases where the Apéry set's nonzero elements form an arithmetic sequence.

Contribution

It characterizes numerical semigroups with gaps equidistributed modulo m, particularly those with an arithmetic progression in their Apéry set.

Findings

Explicit description of such numerical semigroups

Conditions for gaps to be equidistributed modulo m

Analysis of Apéry sets forming arithmetic sequences

Abstract

For a positive integer $m$, a finite set of integers is said to be equidistributed modulo $m$ if the set contains an equal number of elements in each congruence class modulo $m$. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup $S$ is equidistributed modulo $m$. Of particular interest is the case when the nonzero elements of an Ap\'ery set of $S$ form an arithmetic sequence. We explicitly describe such numerical semigroups $S$ and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo $m$.