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Large sum-free sets in finite vector spaces II

This paper resolves a question posed by Leo Versteegen by proving that for $n \ge 3$ , any sum-free set in $\mathbb{F}_5^n$ with size at least $28 \cdot 5^{n-3}$ must either be contained in the union of two parallel hyperplanes or be isomorphic to the product of a specific 28-element sum-free set in $\mathbb{F}_5^3$ and the remaining vector space dimensions.

Original authors: Christian Reiher, Sofia Zotova

Published 2026-04-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Christian Reiher, Sofia Zotova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are organizing a massive party in a multi-dimensional room. The room is built on a grid system, like a giant 3D (or even 10D) version of a Sudoku board, but instead of numbers 1-9, the coordinates are numbers from 0 to 4 (specifically, the field of integers modulo 5).

In this party, there is a strict rule: No three guests can form a "sum trio."
If Guest A and Guest B are at the party, then Guest C (who is exactly the mathematical sum of A and B) is forbidden from attending. A group of people who follow this rule is called a sum-free set.

The Big Question

Mathematicians have long known how to fill this room with the maximum number of guests possible without breaking the rule. It turns out the best way to do this is to pick a giant "wall" (a hyperplane) that doesn't touch the center of the room, and invite everyone standing on that wall and its exact opposite wall. This gives you a huge crowd, but it's a very boring, predictable crowd.

The real puzzle, which this paper solves, is: What is the largest "interesting" party you can throw?
By "interesting," the authors mean a party that doesn't look like those boring walls. They want to find the biggest possible group of guests that is sum-free but refuses to fit inside those standard walls.

The Discovery: The "VL-Set"

The authors, Christian Reiher and Sofia Zotova, answer a question posed by a mathematician named Leo Versteegen. They prove that for any room with 3 or more dimensions, there is a specific "ceiling" on how big an interesting party can be.

If you try to throw a party larger than this ceiling, you have two choices:

Give up and make it boring: Your party will inevitably collapse into one of those standard "wall" shapes.
Stick to the secret recipe: Your party must look exactly like a specific, weirdly shaped configuration they call a VL-set (named after Vsevolod Lev and Leo Versteegen).

The Size Limit:
The paper proves that the maximum size for this "interesting" party is exactly 28 times 5 to the power of (n-3).

Think of it like this: If you have a 3D room, the max size is 28.
If you add a 4th dimension, the size jumps by a factor of 5.
If you add a 5th dimension, it jumps by another factor of 5.

The "Fishy" Detective Work

How did they prove this? They didn't just count people; they used a clever mathematical detective technique involving "Fishy Functions."

Imagine you take a photo of your party and project it onto a flat 2D screen. Instead of seeing individual people, you see a heatmap where the brightness of each square represents how many people are standing in that column of the room.

The authors realized that if your party is huge and "interesting," this heatmap has to look very specific. They called these special heatmaps "Fishy." Why "Fishy"? Because they have strange, suspicious properties:

They can't be too bright in any single spot (no one column is too crowded).
The total brightness has to be high enough to count as a big party.
If you look at three spots that form a triangle, their brightnesses can't add up to a weird number (like 6.5); they have to follow strict integer rules.

The authors spent a huge amount of time (the middle sections of the paper) proving that there are only three ways to draw a "Fishy" heatmap that is big enough and doesn't look like a boring wall.

The Alpha Fish: A specific cross-shape pattern.
The Beta Fish: A pattern that fills up most of the room except for a few holes.
The Gamma Fish: Another specific, intricate pattern.

They proved that if your party is big enough, its "heatmap" must look like one of these three fish. And if the heatmap looks like one of these fish, the actual party structure is forced to be the VL-set.

The Analogy of the "Magic Wall"

To visualize the proof, imagine trying to build a tower of blocks that doesn't fall over (the sum-free rule).

The Normal Way: You build a straight, flat wall. It's stable and huge.
The "Interesting" Way: You try to build a tower that leans, twists, and has holes, but is still huge.
The Result: The authors say, "If your tower is taller than 28 blocks (in the base case), you can't twist it anymore. It will either snap back into a straight wall, or it will have to be built exactly like our secret 'VL' blueprint."

Why Does This Matter?

This might sound like a game of abstract math, but it's actually about structure and limits.

Coding Theory: These "sum-free" rules are similar to how we design error-correcting codes for space communication. Knowing the limits of these sets helps engineers build better, more efficient ways to send data without errors.
Mathematical Maturity: This paper closes the final chapter on a specific type of problem that has been open for decades. For every other prime number (2, 3, 7, 11, etc.), mathematicians already knew the answer. The number 5 was the stubborn outlier that refused to give up its secrets. This paper finally cracked the code for the number 5.

In a Nutshell

The paper is a mathematical detective story. The authors set out to find the biggest possible "weird" party in a high-dimensional grid. They proved that if the party gets too big, it loses its "weirdness" and becomes a boring wall, unless it follows a very specific, secret blueprint (the VL-set) discovered by two other mathematicians. They used a method of projecting the party onto a 2D screen and analyzing "Fishy" patterns to prove that no other shapes are possible.

It's a victory for order over chaos, showing that even in the infinite possibilities of high-dimensional math, there are strict, beautiful rules that govern how things can fit together.

1. Problem Statement and Context

The Core Problem:
The paper addresses the classification of large sum-free sets in finite vector spaces over the field $\mathbb{F}_5$ . A subset $A$ of an abelian group $G$ is sum-free if there are no solutions to $x + y = z$ with $x, y, z \in A$ .

The authors focus on the invariant $sf_1(G)$ , defined as the maximum cardinality of a sum-free set $A \subseteq G$ that is not contained in any maximum sum-free set (i.e., $A \notin \mathcal{SF}_0(G)$ , where $\mathcal{SF}_0(G)$ is the collection of all maximum sum-free sets).

Background:

For $G = \mathbb{F}_p^n$ , the maximum size $sf_0(\mathbb{F}_p^n)$ is known. For $p=5$ , $sf_0(\mathbb{F}_5^n) = 2 \cdot 5^{n-1}$ . The maximum sets are of the form $H \cup (-H)$ , where $H$ is an affine hyperplane not containing the origin. These are called normal sets.
The problem of determining $sf_1(\mathbb{F}_p^n)$ was solved for $p=2$ (Davydov and Tombak), $p=3$ (Lev), and all primes $p \ge 7$ (Lev and Versteegen).
The Open Case: The case $p=5$ remained open. While upper bounds existed ( $sf_1(\mathbb{F}_5^n) < 1.5 \cdot 5^{n-1}$ ), the exact value and the structure of extremal sets were unknown for $n \ge 3$ .

The Specific Question:
Leo Versteegen had previously identified a specific construction (the VL-set) of size $28 \cdot 5^{n-3}$ for $n \ge 3$ . The paper aims to prove that this construction is optimal and to classify all sum-free sets of this size.

2. Methodology

The proof is a sophisticated combination of combinatorial number theory, structural analysis, and linear algebra. The methodology is broken down into three main pillars:

A. Kneser's Theorem and Sum-Set Estimates

The authors heavily utilize Kneser's Theorem, which provides a lower bound on the size of the sumset $A+B$ in an abelian group based on the symmetry subgroup $K = \text{Symp}(A+B)$ .

They use this to derive constraints on the distribution of elements in sum-free sets.
Lemma 5.2 is a critical technical tool: It states that if three non-empty sets $X, Y, Z$ satisfy $(X+Y) \cap Z = \emptyset$ and have sufficiently large sizes relative to the dimension, they must be covered by a small number of translates of a hyperplane. This allows the authors to reduce high-dimensional problems to lower-dimensional ones.

B. The "Fishy" Function Framework

To analyze the density of sum-free sets, the authors introduce a projection method. For a sum-free set $A \subseteq \mathbb{F}_5^n$ and a linear epimorphism $\phi: \mathbb{F}_5^n \to \mathbb{F}_5^2$ , they define a function $f_A^\phi: \mathbb{F}_5^2 \to \mathbb{R}_{\ge 0}$ where:
$f_A^\phi(x) = \frac{|A \cap \phi^{-1}(x)|}{5^{n-3}}$
They define a function $f$ as "fishy" if it satisfies three specific properties derived from the sum-free condition:

Norm Constraints: $\|f\|_1 \ge 28$ and $\|f\|_\infty \le 5$ .
Parity Constraint: No subset sum equals a half-integer (related to the integrality of set sizes).
Sum-Free Inequality: For any $P, Q$ and $\epsilon \in \{-1, 1\}$ , if $f(P), f(Q), f(P+\epsilon Q) > 0$ , then $f(P) + f(Q) + \lfloor f(P+\epsilon Q) \rfloor \le 6$ .

The core of the proof involves classifying all "fishy" functions. They prove that if a fishy function has a maximum value $>3$ , its support is either covered by three parallel lines (implying the set is normal) or it is isomorphic to one of three specific extremal functions ( $f_\alpha, f_\beta, f_\gamma$ ).

C. Inductive Structure and "Acceptable" Functions

For the case where the projection to $\mathbb{F}_5^2$ does not immediately yield a normal set, the authors introduce "acceptable" functions on $\mathbb{F}_5^3$ . These functions satisfy stricter conditions related to the existence of a "special point" $P^\star$ with high density.

Proposition 2.9 proves that for any acceptable function, there exists a line $L$ in $\mathbb{F}_5^3$ such that the sum of values on $L$ is $>3$ .
This result is used to lift the structural properties from dimension 3 to dimension $n$ via induction.

3. Key Contributions and Results

Main Theorem (Theorem 1.3)

The paper establishes the exact value of the invariant and classifies the extremal sets:

Theorem: For every $n \ge 3$ , $sf_1(\mathbb{F}_5^n) = 28 \cdot 5^{n-3}$ .
Furthermore, every sum-free set $A \subseteq \mathbb{F}_5^n$ with $|A| \ge 28 \cdot 5^{n-3}$ is either:

Normal: Contained in $H \cup (-H)$ for some affine hyperplane $H$ not containing the origin.

A VL-set: Isomorphic to $\Lambda \times \mathbb{F}_5^{n-3}$ , where $\Lambda \subset \mathbb{F}_5^3$ is a specific sum-free set of size 28.

The VL-Set Construction

The set $\Lambda$ (discovered by Lev and Versteegen) is defined as:
$\Lambda = \{(1,0), (-1,0), (0,1), (0,-1)\} \times \mathbb{F}_5 \cup \{(-1,2,0), (-1,2,1), (2,1,1), (2,1,2), (1,-2,0), (1,-2,-1), (-2,-1,-1), (-2,-1,-2)\}$
This set is sum-free, non-normal, and has size 28.

Classification of Fishy Functions (Proposition 2.6)

A major technical contribution is the complete classification of "fishy" functions $f: \mathbb{F}_5^2 \to \mathbb{R}_{\ge 0}$ with $\|f\|_\infty > 3$ . The authors prove that such a function is either:

Supported on the union of three parallel lines (leading to normal sets).
Isomorphic to one of three specific configurations ( $f_\alpha, f_\beta, f_\gamma$ ) depicted in the paper.
The analysis of these three configurations leads directly to the identification of the VL-sets.

Base Case Classification (Theorem 1.1)

The authors provide a complete classification of non-normal sum-free sets in $\mathbb{F}_5^2$ with size $\ge 5$ , showing there are only two isomorphism classes. This serves as the base case for the induction.

4. Significance

Resolution of a Long-Standing Open Problem: This paper closes the final open case for the determination of $sf_1(\mathbb{F}_p^n)$ for all primes $p$ . It completes the picture for vector spaces over finite fields regarding the second-largest sum-free sets.
Structural Insight: The work demonstrates that for $p=5$ , the "second best" sum-free sets are not random or chaotic but are highly structured, arising from a specific 3-dimensional construction ( $\Lambda$ ) extended trivially to higher dimensions.
Methodological Advancement: The introduction of "fishy" functions and the rigorous application of Kneser's theorem to bound the support of these functions provide a powerful new toolkit for extremal combinatorics in finite abelian groups.
Connection to Aperiodic Sets: The paper briefly touches on the invariant $tp(G)$ (the size of the largest aperiodic maximal sum-free set). The result implies $tp(\mathbb{F}_5^3) = 28$ , while $tp(\mathbb{F}_5^4)$ remains an open problem, suggesting a new direction for future research.

In summary, Reiher and Zotova provide a definitive answer to the structure of large sum-free sets in $\mathbb{F}_5^n$ , proving that the construction by Lev and Versteegen is optimal and unique up to isomorphism for the non-normal case.