Determine the maximum size of aperiodic maximal sum-free subsets in F_5^n for n ≥ 4

Determine t(F_5^n) for every integer n ≥ 4, where t(G) denotes the maximum size of an aperiodic, inclusion-wise maximal sum-free subset of the finite abelian group G (with Sym(X) = {0} for aperiodic X), and t(G) = 0 if no such subset exists.

Background

The paper introduces the invariant t(G): a subset X of an abelian group G is called aperiodic if Sym(X) = {0}. If there exists an aperiodic, inclusion-wise maximal sum-free subset of G, then t(G) is the largest size such a set can have; otherwise, t(G) is defined to be 0.

For vector spaces over finite fields with prime p ≠ 5, the values t(F_p^n) are known. In the case p = 5, only the small dimensions are settled: t(F_5) = 0, t(F_5^2) = 5, and t(F_5^3) = 28 (the last coming from the aperiodic set Λ constructed in this work). The authors explicitly state that determining t(F_5^n) for higher dimensions remains open.