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

Abstract

For a prime $p\equiv 2(\mathrm{mod}3)$, 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 $\{\frac{p+1}{3},\frac{p+1}{3}+1,\dots ,\frac{2p-1}{3}\}\times {\mathbb{F}}_p^{n-1}$ 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 $p\equiv \mu (\mathrm{mod}k+\ell )$ with $2\le \mu \le k+\ell -1$, and show that the extremal configurations are precisely $\lceil (\mu -1)/2\rceil$ 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 $2\le \mu \le k+\ell -3$ (which is tight), only two structural types occur for all $k+\ell \ge 5$; and when $\mu =2$ 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.

40 pages, 0 figure