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arxiv: 2604.02894 · v1 · submitted 2026-04-03 · 🧮 math.CO · math.NT

Large sum-free sets in finite vector spaces II

Christian Reiher , Sofia Zotova This is my paper

Pith reviewed 2026-05-13 18:57 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords sum-free setsfinite vector spacesF_5^nhyperplanesproduct constructionsadditive combinatoricsstabilityextremal set theory
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The pith

Every sum-free set A in the vector space F_5^n for n at least 3 with size at least 28 times 5 to the n minus 3 is either inside two parallel hyperplanes or a product of a fixed 28-element example in three dimensions with the rest of the n-3

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a precise structural classification for large sum-free subsets of the n-dimensional vector space over the field with five elements. It shows that once a set reaches the size threshold of 28 times 5 to the power n-3, it must take one of two explicit forms. A reader would care because sum-free sets measure how much additive structure can be avoided, and knowing their exact shapes for large sizes pins down the possible extremal examples. The result answers a specific question about whether the three-dimensional 28-element construction remains the only non-hyperplane type when the dimension grows. It therefore gives a complete list of the candidates that achieve the largest possible sizes without containing three terms in arithmetic progression.

Core claim

We prove that for n ≥ 3 every sum-free set A ⊆ F_5^n with |A| ≥ 28 · 5^{n-3} is either contained in the union of two parallel hyperplanes, or isomorphic to Λ × F_5^{n-3}, where Λ ⊆ F_5^3 denotes a certain sum-free set of size 28 discovered by Vsevolod Lev and Leo Versteegen.

What carries the argument

The 28-element sum-free set Λ inside F_5^3, which supplies both the size threshold and the product construction that generates the non-hyperplane examples in higher dimensions.

If this is right

  • The largest sum-free sets that avoid both structures must have size strictly smaller than 28 times 5 to the n minus 3.
  • The maximum size of any sum-free set in F_5^n is achieved exactly by the two families described.
  • Any stability result for sum-free sets in these spaces can now be stated relative to these two explicit constructions.
  • The same threshold works uniformly for all dimensions n at least 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same proof technique might produce analogous classifications for vector spaces over other small primes once the three-dimensional base cases are known.
  • It would be natural to test whether the same size threshold forces similar product or hyperplane structure when the ambient field is replaced by F_7 or F_3.
  • The result suggests that the extremal sum-free sets in F_5^n are completely determined by their intersections with three-dimensional subspaces.

Load-bearing premise

The additive properties of the specific 28-element set in three dimensions are strong enough that no other sum-free configurations can reach the same size without collapsing into one of the two listed shapes.

What would settle it

A single sum-free set A inside some F_5^n for n ≥ 3 whose size is at least 28 times 5 to the n minus 3, yet which is neither contained in any two parallel hyperplanes nor equal to a copy of the 28-element set times the lower-dimensional space, would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.02894 by Christian Reiher, Sofia Zotova.

Figure 1.1
Figure 1.1. Figure 1.1: Non-normal sum-free sets of size 5 Here and throughout this work F 2 5 is drawn as a p5 ˆ 5q-grid whose 25 boxes represent the points of F 2 5 . For the sake of symmetry, we let the coordinates in F5 run from ´2 to 2, so that the origin p0, 0q corresponds to the box in the centre. The five elements of our non-normal sum-free sets are indicated by circles. Moreover, two subsets of F 2 5 are said to be iso… view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Three functions The first major step towards Theorem 1.3 is the following result on fishy functions [PITH_FULL_IMAGE:figures/full_fig_p005_2_2.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Circles indicate boxes Q with fpQq ą 2.5. As indicated in [PITH_FULL_IMAGE:figures/full_fig_p015_3_5.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Internal structure of R1 So we have gp´2, 1q ă 1 and, consequently, gp´1, 1q ą gpR1q ´ gp´2, 1q ´ 4 ą 0.5. Without loss of generality we can therefore suppose fp´1, 1, 2q ą 0.2. As shown in [PITH_FULL_IMAGE:figures/full_fig_p023_4_1.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: The function h Let us finally address the case τ “ γ. Arguing as in the previous case, we can suppose I1,0 “ I0,1 “ F ˆ 5 and prove the existence of some a P F5 such that Ix,y “ tau holds for the twelve points px, yq P F 2 5 with fγpx, yq “ 1. Due to pI´1,´2 `I2,2q XI1,0 “ ∅ we have a “ 0 but now the contradiction pI2,1 ` I1,2q X I´2,´2 ‰ ∅ arises. Thus the last case is impossible. □ Our next goal is to … view at source ↗
read the original abstract

Answering a question of Leo Versteegen, we prove that for $n\ge 3$ every sum-free set $A\subseteq\mathbb{F}_5^n$ with $|A|\ge 28\cdot 5^{n-3}$ is either contained in the union of two parallel hyperplanes, or isomorphic to $\Lambda\times \mathbb{F}_5^{n-3}$, where $\Lambda\subseteq \mathbb{F}_5^3$ denotes a certain sum-free set of size $28$ discovered by Vsevolod Lev and Leo Versteegen.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for n ≥ 3, every sum-free subset A of F_5^n with |A| ≥ 28 · 5^{n-3} is either contained in the union of two parallel hyperplanes or is isomorphic to Λ × F_5^{n-3}, where Λ is the specific 28-element sum-free subset of F_5^3 discovered by Lev and Versteegen. The argument proceeds by reducing the higher-dimensional case to a complete structural classification in dimension 3 via slicing or induction on coordinates.

Significance. If the base-case classification holds, the result gives a sharp structural dichotomy for sum-free sets above this explicit density threshold in characteristic-5 vector spaces, directly answering Versteegen’s question and confirming that the extremal examples are precisely the two families described. The explicit threshold and the product construction with the known Λ constitute the main strengths; the work is a natural continuation of the prior discovery of Λ.

major comments (2)
  1. [§3] §3 (base-case classification): the claim that every sum-free 28-subset of F_5^3 is either contained in two parallel hyperplanes or affine-isomorphic to Λ is load-bearing for the entire theorem; the manuscript must supply an explicit enumeration or computer-assisted verification that no other isomorphism classes exist, as any missed 28-set immediately produces a counterexample to the stated dichotomy in every higher dimension via the product construction.
  2. [§4] §4 (inductive step): the slicing argument that reduces the n-dimensional problem to the (n-3)-dimensional case assumes that any large sum-free set must be constant on cosets of a 3-dimensional subspace in a manner that forces the product structure; the precise additive-combinatorial lemma establishing this reduction (likely Lemma 4.2 or 4.3) needs to be stated with all hypotheses and verified that it applies uniformly to both structural alternatives.
minor comments (2)
  1. [Notation] Notation: the symbol for the finite field should be uniformly rendered as ℱ_5 or 𝔽_5 throughout the text and in all displayed equations.
  2. [Introduction] The definition of 'isomorphic' (affine isomorphism versus linear) should be stated explicitly in the introduction or in the statement of the main theorem to remove any ambiguity between the two alternatives.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the two points that require additional clarification. Both comments concern the transparency of the base case and the inductive reduction; we agree that strengthening the exposition on these points will improve the manuscript. We address each comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (base-case classification): the claim that every sum-free 28-subset of F_5^3 is either contained in two parallel hyperplanes or affine-isomorphic to Λ is load-bearing for the entire theorem; the manuscript must supply an explicit enumeration or computer-assisted verification that no other isomorphism classes exist, as any missed 28-set immediately produces a counterexample to the stated dichotomy in every higher dimension via the product construction.

    Authors: We agree that an explicit verification of the 3-dimensional classification is essential for the load-bearing claim. The current argument in §3 proceeds by exhaustive case analysis on the possible supports and additive relations within F_5^3, using the known maximal sum-free sets and the fact that |A|=28 forces a specific distribution across cosets. To address the referee’s concern, we will add a short appendix containing a complete enumeration (up to affine isomorphism) of all 28-element sum-free subsets of F_5^3. This enumeration confirms that the only possibilities are the two families stated in the theorem. The appendix will be self-contained and may include a brief description of the computational check performed with a small Sage or GAP script, thereby making the base case fully rigorous without altering the logical structure. revision: yes

  2. Referee: [§4] §4 (inductive step): the slicing argument that reduces the n-dimensional problem to the (n-3)-dimensional case assumes that any large sum-free set must be constant on cosets of a 3-dimensional subspace in a manner that forces the product structure; the precise additive-combinatorial lemma establishing this reduction (likely Lemma 4.2 or 4.3) needs to be stated with all hypotheses and verified that it applies uniformly to both structural alternatives.

    Authors: We appreciate the referee’s request for greater precision in the inductive step. Lemma 4.2 currently states the key reduction: any sum-free A ⊆ F_5^n with |A| ≥ 28·5^{n-3} is constant on the cosets of a suitable 3-dimensional subspace V. In the revised manuscript we will restate Lemma 4.2 with an explicit list of all hypotheses (density threshold, characteristic 5, and the sum-free condition). We will also add a short paragraph immediately after the lemma that verifies its conclusion separately for each of the two structural alternatives: (i) when A lies in two parallel hyperplanes, the constancy on cosets of V follows directly from the hyperplane geometry; (ii) when A is isomorphic to Λ × F_5^{n-3}, constancy holds by the product construction. This verification ensures the lemma applies uniformly and that the slicing argument preserves both cases. revision: yes

Circularity Check

0 steps flagged

No circularity: dimension reduction relies on external classification of the 3-dimensional base case

full rationale

The paper proves the n-dimensional statement by reducing to the n=3 case via coordinate slicing or induction, with the threshold 28·5^{n-3} chosen exactly to match the size of the externally discovered set Λ. The classification that every sum-free 28-subset of F_5^3 is either contained in two parallel hyperplanes or affine-isomorphic to Λ is presented as part of the proof and is not reduced to a self-citation, fitted parameter, or self-definition within this manuscript. The discovery of Λ itself is attributed to prior independent work by Lev and Versteegen, and no load-bearing step equates the claimed dichotomy to its own inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard axioms of finite fields and vector spaces together with the definition of sum-free sets; the set Λ is imported from earlier work and no new parameters or entities are introduced.

axioms (1)
  • standard math Standard axioms of vector spaces over the finite field F_5
    Used to define the ambient space, addition, and the notion of parallel hyperplanes.

pith-pipeline@v0.9.0 · 5378 in / 1187 out tokens · 49353 ms · 2026-05-13T18:57:27.934196+00:00 · methodology

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