Asymptotic Bounds for Length Density in Numerical Semigroups

Length density δ(n) measures how close the set of factorization lengths of n in a numerical semigroup is to a full interval. We establish explicit lower bounds and convergence rates, proving that δ(n) → 1 as n → ∞ for arbitrary embedding dimension. Our approach combines extremal length estimates with pruning algorithms for factorization trees, providing both structural insight and computational efficiency. A general pruning template is introduced, clarifying the relation between ordered and unordered pruning; we show unordered pruning is sound only in commutative settings or under commutation relations. Extensions to non-commutative and non-cancellative semigroups are analyzed via right ideals and prefix-ACCP conditions. Numerical experiments illustrate stabilization of δ(n) and confirm asymptotic error bounds. These results place pruning as a unifying tool for asymptotics, computation, and structural analysis of factorization invariants in semigroups.