The Hilton-Milner type results of $(k, \ell)$-sum-free sets in $\mathbb F_p^n$
Xin Wei, Xiande Zhang, Gennian Ge
TL;DR
The paper develops a comprehensive Hilton–Milner type theory for (k,ℓ)-sum-free subsets of 𝔽_p^n, establishing the exact extremal size when p=(k+ℓ)m+2+λ and identifying ⌈(λ+1)/2⌉ extremal cuboids that achieve the (m+1)p^{n−1} bound. It proves sharp stability results: any nontrivial near-extremal set has size at most mp^{n−1}, and it classifies the nontrivial extremal structures in several parameter regimes, including explicit abnormal configurations (types 3–5) for certain (k+ℓ,λ) pairs. The argument integrates additive combinatorics with Fourier-analytic techniques and leverages progress towards the 3k−4 conjecture via a carefully defined T(c) parameter to control how large p must be. The results illuminate the structure of large sum-free sets over finite vector spaces, reveal new connections between inverse additive number theory and extremal problems, and lay groundwork for further exploration of Hilton–Milner type phenomena in related groups and parameter ranges.
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.