2025/12/28 by Wei, Xin, Zhang, Xiande, Ge, Gennian
#2020: Primary 11B30 #43A25 #Combinatorics (math.CO) #FOS: Mathematics #Secondary 05D10
paper · doi:10.48550/arxiv.2512.22835
For a prime p ≡ 2 \pmod 3, it is well known that the largest sum-free subsets of \mathbbFpn have size (p+1)/(3) pn-1, and the extremal sets must be a cuboid of the form \(p+1)/(3), (p+1)/(3)+1, …, (2p-1)/(3)\ × \mathbbFpn-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 (p-2)/(3) pn-1, and all possible structures attaining this bound were classified. In this paper, we develop a general Hilton--Milner theory for (k,ℓ)-sum-free sets in \mathbbFpn for k > ℓ ≥ 1. We determine the maximum size of such sets for all p ≡ μ\pmodk+ℓ with 2 ≤ μ≤ k+ℓ-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,ℓ)-sum-free set not contained in any of these extremal cuboids is uniformly bounded away from the maximum by a gap of order pn-1, and we determine the full structure of all sets achieving this second-best bound in several broad parameter ranges. In particular, when 2 ≤ μ≤ k+ℓ-3 (which is tight), only two structural types occur for all k+ℓ ≥ 5; and when μ= 2 or 3, we obtain a complete classification for all k > ℓ ≥ 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.