Attainment of the Freeze–Schmid bound for rank-three elementary abelian groups

Determine for which primes p the Freeze–Schmid lower bound D_k(C_p^3)\ge 3p-2+3\lfloor p/2\rfloor+\delta+(k-2)p is attained for every k\ge2, and determine the threshold in k beyond which it is attained for the remaining primes.

Background

The paper establishes that the Freeze–Schmid lower bound is attained for every k\ge2 when the group is C_53, while previously known results show attainment for C_23 and failure from k=3 onward for C_33. These examples indicate that the behavior depends on the prime p and may involve a prime-dependent threshold in k.

The unresolved problem is to classify all primes p for which equality holds for every k\ge2 and, when equality does not hold from the outset, to identify the least value of k from which the bound becomes attained. Here \delta=1 for odd p and \delta=0 for p=2, as specified in the displayed Freeze–Schmid bound.

References

For which primes $p$ the bound eq:FS, namely $D_k(C_p3)\ge3p-2+3\lfloor p/2\rfloor+\delta+(k-2)p$, is attained for all $k\ge2$, and where the threshold in $k$ sits otherwise, we do not know.

eq:FS:

Dk(G) ≥ D(G)+s⌊nt2⌋+δ+(k−2)nr(k≥2),D_k(G)\ \ge\ D(G)+s\Bigl\lfloor\frac{n_t}{2}\Bigr\rfloor+\delta+(k-2)n_r\qquad(k\ge2),

— The fourth generalized Davenport constant of $C_5^3$  (2609.04950 - Yiu, 4 Sep 2026) in Remark 2.14, Section 6.1 (Relation to prior work and scope of the claim)