Asymptotics of extremal p-lengths for singular arithmetical congruence monoids

Determine the asymptotic behavior, as n grows, of the extremal p-length functions e_0(x^n), e_∞(x^n), and l_∞(x^n) for a fixed element x > 1 in a singular arithmetical congruence monoid M_{a,b} = {1} ∪ {n ∈ Z_{>1} : n ≡ a (mod b)} with a^2 ≡ a (mod b) and a ≠ 1, where for a factorization x^n = ∏ u_i^{z_i} into atoms u_i of M_{a,b}, l_0(z) = ∑_i 1_{z_i>0} is the number of distinct atoms used, l_∞(z) = max_i z_i, e_p(x^n) = max over all factorizations of l_p, and l_p(x^n) = min over all factorizations of l_p.

Background

Section 3 studies p-length extremals for arithmetical congruence monoids M_{a,b} = {1} ∪ {n > 1 : n ≡ a (mod b)} with a^2 ≡ a (mod b). The authors define for each factorization of x into atoms u_i with exponents z_i the p-length l_p by l_p(z) = ∑ z_i^p for p ∈ Z_{≥0} and l_∞(z) = max_i z_i, and then define e_p(x) and l_p(x) as the maximal and minimal p-lengths over all factorizations of x, respectively.

For regular ACMs (a = 1), known structural properties imply that, for fixed x > 1, only finitely many atoms divide powers x^n; this yields linear or bounded asymptotics for several extremal functions. In contrast, for singular ACMs (a ≠ 1), the paper shows more subtle behavior: e.g., in M_{4,6} the growth rate of l_∞(x^n) can vary with x, ranging from Θ(n^{1/2}) to Θ(n^{2/3}). These examples motivate a general classification of asymptotics for e_0(x^n), e_∞(x^n), and l_∞(x^n) in the singular case.