On evenly distributed sets of gaps of numerical semigroups

Caleb McKinley Shor

Published 2021-01-05Version 1

For a positive integer $m$, a finite set of integers is said to be evenly distributed 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 evenly distributed modulo $m$. Of particular interest is the case when the nonzero elements of an Ap\'ery set of $S$ form an arithmetic sequence, which occurs precisely when $S$ is a numerical semigroup of embedding dimension 2 or a numerical semigroup of maximal embedding dimension generated by a generalized arithmetic sequence. We derive explicit conditions for which the gaps of these numerical semigroups are evenly distributed modulo $m$.