Close the gap for polylogarithmic macro density in A_n by proving a matching lower bound on expansion

Establish a lower bound on the expansion function f_{G'}(s) for the free abelian monoid A_n when the macro set M has polylogarithmic growth |M ∩ B_G(r)| ≤ c (log(e + r))^q, ideally matching the known quasi-exponential upper bound exp(K s log s) to determine the precise asymptotic expansion rate in this regime.

Background

For A_n, the authors prove that a macro set with polylogarithmic density yields at most quasi-exponential expansion, providing an upper bound of the form exp(K s log s). However, no corresponding lower bound is established, leaving the true asymptotic rate unresolved.

This regime lies between logarithmic-density macros (which give exponential expansion) and polynomial-density macros (which can yield infinite expansion), making the exact behavior of polylogarithmically dense macro sets a key missing piece in the overall picture.