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A proof of the Freiman-Lev conjecture

This paper presents a complete proof of the long-standing Freiman-Lev conjecture regarding restricted sumsets by resolving its final and most challenging open case for sets of integers where the two largest elements satisfy specific lower bounds.

Original authors: Yujie Wang, Min Tang

Published 2026-08-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yujie Wang, Min Tang

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 a detective trying to solve a mystery about how numbers mix together. In the world of mathematics, there is a branch called "additive number theory," which is essentially the study of what happens when you take a group of numbers and start adding them up. If you have a set of numbers, say $\{1, 3, 5\}$ , and you add every possible pair, you get a new collection of sums: $\{1+3, 1+5, 3+5\} = \{4, 6, 8\}$ . Mathematicians call this new collection a "sumset."

But here is the twist: what if you are only allowed to add different numbers together? You can't add a number to itself (so no $1+1$ or $3+3$ ). This creates a "restricted sumset." It's like a party where everyone must dance with a partner, but no one is allowed to dance with themselves. The big question mathematicians have been asking for decades is: "If I start with a specific number of guests (integers), how many unique dance pairs (sums) can I guarantee will happen?"

For a long time, there was a famous rule for the standard "dance" (where self-dancing is allowed), but the "no self-dancing" version was much trickier. It turned out that the structure of the original group of numbers matters a lot. If the numbers are packed tightly together, you get fewer unique sums. If they are spread out, you get more. For years, mathematicians had a very strong guess—a "conjecture"—about the absolute minimum number of unique sums you could get, no matter how you arranged your numbers, as long as they followed certain basic rules (like having no common divisor other than 1). This guess was known as the Freiman-Lev conjecture. It was like having a map that showed the lowest possible valley in a mountain range, but there was one tiny, foggy peak where no one could be sure if the valley went any lower.

This paper is the final piece of the puzzle. The authors, Yujie Wang and Min Tang, have successfully climbed that last foggy peak and proved that the Freiman-Lev conjecture is absolutely true. They didn't just guess or simulate; they built a rigorous mathematical proof that leaves no room for doubt.

The Story of the Proof

To understand what the authors did, imagine you have a set of $k$ integers, which we'll call our "guest list." Let's say the smallest guest is 0 and the largest is some big number $l$ . The authors are interested in the "restricted sumset," which is the collection of all sums you can make by adding two different guests from the list.

For a long time, mathematicians knew that if the guest list is "dense" (the numbers are close together), the number of sums is relatively small. But if the list is "sparse" (the numbers are far apart), the number of sums grows. The Freiman-Lev conjecture proposed a specific formula for the minimum number of sums you can get, depending on how spread out the largest numbers are.

The formula says:

If the numbers are packed tightly (specifically, if the largest number $l$ is less than or equal to $2k - 5$ ), the number of sums is at least $l + k - 2$ .
If the numbers are more spread out (if $l$ is at least $2k - 4$ ), the number of sums is at least $3k - 7$ .

The tricky part was the second case. For years, mathematicians could prove this lower bound for almost every situation, but there was a specific, stubborn scenario where the math got messy. This happened when the second-to-last number in the list ( $a_{k-2}$ ) was at least $2k - 4$ and the very last number ( $a_{k-1}$ ) was at least $2k - 2$ . It was like trying to solve a jigsaw puzzle where you had all the pieces except for the one that fit in the very center.

Wang and Tang's paper, titled "A proof of the Freiman-Lev conjecture," tackles this final, most challenging case. They didn't just look at the numbers; they analyzed the "shape" of the set. They used a clever strategy involving "gap sets" (numbers that are missing from the list) and "locally dense sets" (groups where numbers are packed tightly in the beginning).

The authors broke the problem down into smaller, manageable chunks using a method called "induction." Think of it like climbing a ladder: if you can prove the rule works for a small number of guests, and you can prove that if it works for $k$ guests, it must also work for $k+1$ guests, then it works for everyone. However, the ladder had a few broken rungs in the middle. The authors had to invent new "combinatorial lemmas" (which are like specialized tools or rules of logic) to fix those rungs.

They examined specific patterns, such as when the numbers in the set follow a rule like $a_i < 2i$ (meaning the $i$ -th number is less than twice its position). They showed that even in these complex, "locally dense" situations, the number of sums never drops below the magic number $3k - 7$ . They also looked at what happens when you take a dense group of numbers and add a few very large numbers to the end of the list. They proved that adding these large numbers forces the number of sums to jump up, ensuring the minimum limit is never breached.

By combining these structural insights with careful logical arguments, they demonstrated that no matter how you arrange your integers (as long as they meet the basic criteria), you cannot create a scenario where the number of unique sums is less than $3k - 7$ when the numbers are spread out enough.

The Conclusion

The paper concludes with a definitive statement: The Freiman-Lev conjecture is true. The authors have resolved the final, most difficult case where the second-to-last and last numbers are large. This means the mathematical community now has a complete and proven answer to the question of how many sums you can guarantee from a set of integers when you forbid adding a number to itself.

There are no "maybe" or "likely" statements here. The authors have provided a complete proof. They didn't just suggest a pattern; they showed that any attempt to break the rule leads to a logical contradiction. The mystery of the restricted sumset's minimum size is officially solved, closing the book on a problem that had puzzled mathematicians for decades. The "foggy peak" has been cleared, and the map is now complete.

Technical Summary: A Proof of the Freiman-Lev Conjecture

Problem Statement
The paper addresses the Freiman-Lev conjecture, a central problem in additive number theory concerning the lower bound of the cardinality of restricted sumsets. Let $A = \{a_0, a_1, \dots, a_{k-1}\}$ be a set of $k > 7$ integers with $0 = a_0 < a_1 < \dots < a_{k-1}$ and $\gcd(A) = 1$ . The restricted sumset is defined as $2^\wedge A = \{a + b : a, b \in A, a \neq b\}$ .

While the conjecture was known to hold for various ranges of the diameter $l = a_{k-1}$ , it remained an open problem specifically in the case where the second largest element satisfies $a_{k-2} \ge 2k - 4$ and the largest element satisfies $a_{k-1} \ge 2k - 2$ . The conjecture posits that under these conditions:
$|2^\wedge A| \ge 3k - 7.$

Methodology
The authors employ a rigorous combinatorial and inductive approach, building upon a series of structural lemmas and inverse theorems developed in their previous works [9, 10]. The proof strategy involves:

Structural Classification and Gap Analysis: The paper defines a "gap set" $W$ consisting of integers in $[0, a_{k-1}] \setminus A$ whose sums with $0$ or $a_{k-1}$ are absent from $2^\wedge A$ . The authors utilize pigeonhole principles to characterize the distribution of elements in $A$ when $W$ is non-empty, particularly in critical diameter ranges ( $a_{k-1} = 2k-3$ or $2k-4$ ).
Locally Dense Sets: A significant portion of the analysis focuses on "locally dense" sets where $a_i < 2i$ for $1 \le i \le k-2$ . The authors establish that specific initial configurations (e.g., $(a_2, a_3, a_4) = (3, 5, 6)$ ) force the restricted sumset size to exceed the conjectured lower bound ( $|2^\wedge A| \ge 3k - 5$ ).
Translation and Recursive Generation: The proof utilizes translation operations (e.g., $B = A \setminus \{a_0\} - a_1$ ) and recursive generation relations to transform complex internal additive conditions into tractable subproblems. This allows the authors to relate the structure of $A$ to smaller sets or reflected sets $A^*$ .
Inductive Framework: The main proof (Theorem 1.1) proceeds by induction on $k$ . The base cases ( $3 \le k \le 9$ ) are verified computationally and structurally. For $k \ge 10$ , the authors decompose the set $A$ into subsets (e.g., $A_1, A_2$ or $B_1, B_2$ ) based on the distribution of elements relative to the threshold $2i$ . They analyze the intersection of sumsets of these subsets to derive global lower bounds.
Case Analysis: The proof systematically eliminates potential counterexamples by analyzing specific structural cases, including:
- Sets with large greatest common divisors in translated subsets.
- Sets where elements satisfy specific additive generation patterns (e.g., $a_s \in 2^\wedge(A \setminus \{a_0\})$ ).
- Extremal sets attaining the bound $3k-7$ , for which complete characterizations are provided.

Key Contributions and Results

Resolution of the Final Case: The paper provides a complete proof of the Freiman-Lev conjecture for the previously open case where $a_{k-2} \ge 2k - 4$ and $a_{k-1} \ge 2k - 2$ .
Theorem 1.1: The main result states that for any set $A$ of $k > 7$ integers with $0=a_0 < \dots < a_{k-1}$ , $\gcd(A)=1$ , $a_{k-2} \ge 2k-4$ , and $a_{k-1} \ge 2k-2$ , the inequality $|2^\wedge A| \ge 3k - 7$ holds.
Structural Characterizations: The authors provide detailed inverse results, characterizing the specific structures of sets that achieve the lower bound $|2^\wedge A| = 3k - 7$ . These extremal sets are shown to be highly regular, often forming unions of arithmetic progressions with common differences modulo 3.
Extension of Previous Work: The paper integrates and extends results from Theorems E and G (from prior works [9, 10]), which covered the ranges $2k-4 \le l \le 2k-3$ and $a_{k-2} < 2k-4$ , respectively, to cover the entire domain of the conjecture.

Significance
The paper claims to complete the proof of the Freiman-Lev conjecture, a well-known problem in the field of restricted sumsets. By resolving the final and most challenging case, the authors establish a definitive lower bound for the cardinality of restricted sumsets for integers with coprime elements. The work relies on and synthesizes a complex framework of combinatorial lemmas, gap set analysis, and inductive arguments to demonstrate that the bound $3k-7$ is tight and universally applicable under the stated conditions. The characterization of extremal sets further contributes to the understanding of the structural properties of sets with small restricted sumsets.

The Story of the Proof

The Conclusion

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