Apéry sets of shifted numerical monoids

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of “shifted” monoids Mn obtained by adding n to each generator of S. In this paper, we characterize the Apéry set of Mn in terms of the Apéry set of the base monoid S when n is sufficiently large. We give a highly efficient algorithm for computing the Apéry set of Mn in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of n.