Apéry set with respect to a set
Definition
Let \(C\) be a subset of \(\mathbb{N}\) and \(x \in \langle C \rangle \setminus \{0\}\). It is defined the Apéry set of \(x\) in \(C\), denoted by \(Ap(C,x)\), as
\[ Ap(C,x) = \{s \in \langle C \rangle ~ | ~ s - x \not \in \langle C \rangle \}. \]
This definition comes from the concept of Apéry set of an element in a numerical semigroup.
Examples
\(\circ\) Let \(C = \{5, 7, 13, 16 \}\), \(C' := \langle C \rangle = \langle 5, 7, 13, 16 \rangle = \{0, 5, 7, 10, 12, \rightarrow \}\) and \(x = 5\). If \(s \ge 17\), it holds \(s - x \in C'\), then \(Ap(C, 5) \subseteq B\), where \(B = \{0, 5, 7, 10, 12, 13, 14, 15, 16\}\). If we check the condition on each element of \(B\), it is obtained that \(Ap(C, 5) = \{0, 7, 13, 14, 16\}\). As \(gcd(C) = 1\), \(C'\) is a numerical semigroup and the definition of Apéry set of a set of integers and the Apéry set of a numerical semigroup are the same.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(C\) be a subset of \(\mathbb{N}\) such that \(gcd(C) = 1\) and \(x \in \langle C \rangle \setminus \{0\}\), the function NumericalSemigroup generates the numerical semigroup generated by \(C\). On the other hand, the functionAperyList returns the Apéry set of \(x\) in \(C\).
gap> C := NumericalSemigroup(9, 24, 31, 35);
<Numerical semigroup with 4 generators>
gap> AperyList(C, 9);
[ 0, 55, 83, 48, 31, 59, 24, 70, 35 ]References
https://gap-packages.github.io/ numericalsgps.