Equidistribution of numerical semigroup gaps modulo $m

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 $m$.