The Hilton-Milner type results of $(k, \ell)$-sum-free sets in $\mathbb F_p^n$

Abstract: For a prime $p \equiv 2 \pmod 3$, it is well known that the largest sum-free subsets of $\mathbb{F}_p^n$ have size $\frac{p+1}{3} p^{n-1}$, and the extremal sets must be a cuboid of the form ${\frac{p+1}{3}, \frac{p+1}{3}+1, \ldots, \frac{2p-1}{3}} \times \mathbb{F}_p^{n-1}$ up to isomorphism. Recently, Reiner and Zotova proved a Hilton--Milner type stability result showing that for large $p$, any sum-free set not contained in the extremal cuboid has size at most $\frac{p-2}{3} p^{n-1}$, and all possible structures attaining this bound were classified. In this paper, we develop a general Hilton--Milner theory for $(k,\ell)$-sum-free sets in $\mathbb{F}_p^n$ for $k > \ell \ge 1$. We determine the maximum size of such sets for all $p \equiv μ\pmod{k+\ell}$ with $2 \le μ\le k+\ell-1$, and show that the extremal configurations are precisely $\lceil (μ-1)/2 \rceil$ non-isomorphic cuboids. Beyond the extremal regime, we prove sharp Hilton--Milner type stability results showing that, for all sufficiently large $p$, a $(k,\ell)$-sum-free set not contained in any of these extremal cuboids is uniformly bounded away from the maximum by a gap of order $p^{n-1}$, and we determine the full structure of all sets achieving this second-best bound in several broad parameter ranges. In particular, when $2 \le μ\le k+\ell-3$ (which is tight), only two structural types occur for all $k+\ell \ge 5$; and when $μ= 2$ or $3$, we obtain a complete classification for all $k > \ell \ge 1$. Our arguments combine additive combinatorics and Fourier-analytic methods, and make use of recent progress toward the long-standing $3k-4$ conjecture, highlighting new connections between inverse additive number theory and extremal problems over finite vector spaces.