Limiting codensity for cyclic groups with k prime factors

Establish that for every positive integer k, the limit of the codensity of the reduced formal context for the lattice of transfer systems on [n1] × ··· × [nk] exists and equals (2^k − 1)/6^k; equivalently, show that lim_{n1,...,nk→∞} ρ(Tr([n1] × ··· × [nk])) = (2^k − 1)/6^k, where Tr([n1] × ··· × [nk]) denotes the lattice of transfer systems on the product lattice [n1] × ··· × [nk] (which is isomorphic to Sub(C_{p1^{n1}···pk^{nk}})).

Background

The paper develops explicit formulas and bounds for the codensity of the reduced formal context associated to the lattice of transfer systems on various families of groups. For cyclic groups, Sub(C_N) is identified with a product of chains [n1] × ··* × [nk], and closed-form codensities are computed for k = 1, 2, and 3, together with limiting values for these cases.

Based on computed values for small k, the authors observe a pattern in the limiting codensities as all exponents ni tend to infinity and propose a general formula for arbitrary k. Confirming this limiting behavior would unify the asymptotic codensity across cyclic groups with any number of prime factors.