Perspicacious $l_p$ norm parameters

Abstract

Fix $t\in [1,\infty ]$. Let $S$ be an atomic commutative semigroup and, for all $x\in S$, let ${\mathcal{L}}_t(S):=\{\Vert f{\Vert }_t:f\in Z(x)\}$ be the "$t$-length set" of $x$ (using the standard $l_p$-space definition of $\Vert \cdot {\Vert }_t$). The $t$-Delta set of $x$ (denoted ${\Delta }_t(S)$) is the set of gaps between consecutive elements of ${\mathcal{L}}_t(S)$; the Delta set of $S$ is then defined by $\bigcup _{x\in S}{\Delta }_t(S)$. Though all existing literature on this topic considers the $1$-Delta set, recent results on the $t$-elasticity of Numerical Semigroups (Behera et. al.) for $t\ne 1$ have brought attention to other invariants, such as the $t$-Delta set for $t\ne 1$, as well. Here we characterize ${\Delta }_t(S)$ for all numerical semigroups $⟨a_1,a_2⟩$ and all $t\in (1,\infty )$ outside a small family of extremal examples. We also determine the cardinality and describe the distribution of that aberrant family.