Growth range of the block size in the wreath-product Erdős–Turán law

Determine how large the block size k may be allowed to grow with n when the base subgroup is the full symmetric group Γ = S_k, while retaining the stated wreath-product Erdős–Turán limit theorem.

Background

Theorem 2.1 establishes a lognormal central limit theorem for the logarithm of the least common multiple of cycle lengths in a uniformly sampled element of Γn ⋊ S_n when k is fixed, for arbitrary Γ ⊆ S_k. For the cyclic subgroup Γ = C_k, the theorem also permits k to grow subject to k = e{o(log{3/2} n)}.

The paper then considers the more difficult case Γ = S_k, where the least common multiple involves the cycle structures of independent uniform permutations in S_k. The authors explicitly state that the permissible growth rate of k in this setting is unknown.

References

It would be interesting to see which subgroups $\Gamma \le S_k$ allow one to take $k$ large with $n$ in the above theorem, and how large. For instance with $\Gamma = S_k$ one has $$ O_n = \lcm(C_1 * \lcm(\pi_1), C_2 * \lcm(\pi_2), \dots) $$ where $\pi_1, \pi_2, \dots$ are iid uniform from $S_k$ and $C_1,C_2,\dots$ are the cycle lengths in a uniform permutation from $S_n$. It's not clear how large $k$ can be taken in this case.

— Cutting a unit square and permuting blocks  (2501.13844 - Tung, 23 Jan 2025) in Section 2, immediately following the proof of Theorem 2.1 (Wreath product Erdős–Turán)