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Sets of unit fractions without two members whose average is a unit fraction

This paper disproves a question posed by Erdős and Graham by proving the existence of a constant $c>0$ such that for all sufficiently large $N$ , there exists a subset of $\{1,\dots,N\}$ with size greater than $cN$ where the average of any two distinct reciprocals is not a unit fraction, thereby establishing the best known lower bounds for sets of unit fractions without non-trivial three-term arithmetic progressions.

Original authors: Will Sawin

Published 2026-07-20

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Will Sawin

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 a world made entirely of fractions, but with a very strict rule: every piece must be a "unit fraction." That means the top number is always 1, like 1/2, 1/3, or 1/100. Mathematicians have spent decades playing with these numbers, asking questions like, "How many of these can we line up before we accidentally create a pattern?" One famous pattern they look for is an arithmetic progression, where three numbers sit evenly spaced, like 1/2, 1/3, and 1/6 (since 1/3 is exactly halfway between 1/2 and 1/6).

The specific puzzle this paper tackles is a bit like a game of "no averages." If you pick two different unit fractions, say 1/a and 1/b, and you calculate their average (the number exactly in the middle of them), the game asks: Can we build a huge collection of these fractions where none of the pairs have an average that is also a unit fraction? For a long time, two legendary mathematicians, Erdős and Graham, wondered if such a collection could be truly massive. They suspected that if you tried to make the collection big enough, you would inevitably be forced to include a pair whose average is also a unit fraction. In other words, they thought the "no-average" rule would force the collection to be tiny compared to the total number of available fractions.

This paper, written by Will Sawin, steps into that arena and delivers a surprising twist. The author proves that Erdős and Graham were wrong. It is possible to build a collection of unit fractions that is surprisingly large—so large that it contains a constant percentage of all available numbers up to a certain point—without ever accidentally creating a pair whose average is a unit fraction. The paper doesn't just guess; it provides a rigorous mathematical construction, a specific recipe for building this giant set, and proves that it works for any sufficiently large number. While the author admits the recipe isn't the absolute most efficient one possible, it is enough to shatter the old belief that such a set must be small.

The Great "No-Average" Heist

Think of the numbers from 1 to a huge number $N$ as a massive crowd of people. Each person holds a sign with a number on it. If you pick two people, say Person $a$ and Person $b$ , they represent the unit fractions $1/a$ and $1/b$ . The "average" of their fractions is a special number. If that average turns out to be a unit fraction (like $1/c$ ), then $a$ and $b$ are "banned" from being in our special club together. The goal is to form the biggest possible club where no two members are banned.

For a long time, the math community thought this club would have to be tiny. They believed that as the crowd grew, the rules would get so strict that you could only keep a vanishingly small percentage of people. But Will Sawin says, "Not so fast!" He shows that you can actually keep a massive chunk of the crowd—specifically, more than a fixed percentage $c$ of everyone, no matter how huge the crowd gets.

How the Magic Trick Works

To pull off this heist, the author doesn't just grab random people from the crowd. He uses a very specific filter, a set of rules that acts like a bouncer at an exclusive club.

First, the bouncer kicks out anyone with "too many small prime factors." Imagine prime numbers as the basic building blocks of all numbers (like 2, 3, 5, 7). The bouncer says, "If your number is built from tiny bricks like 2 or 3, you can't come in." This removes a lot of the crowd, but leaves a healthy number of people who are made of larger, more complex bricks.

Second, the bouncer checks the "complexity" of the numbers. He counts how many prime factors a number has (counting repeats, so $12 = 2 \times 2 \times 3$ has three factors). The rule is that you can't have too many factors compared to what is statistically expected for a number of your size. It's like saying, "If you're a medium-sized number, you can't be made of an absurdly large pile of bricks."

The genius of the paper lies in proving that if you stick to this filtered group, the "bad pairs" (the ones whose average is a unit fraction) become incredibly rare. The author uses a clever mathematical trick involving a change of variables—essentially renaming the numbers to make the pattern easier to see—to show that the average number of "bad pairs" for any single person in this group is very low.

In fact, the math shows that for the vast majority of people in this filtered group, there are almost no partners they can't pair with. By carefully counting these interactions, the author proves that even after removing the few people who do have a forbidden partner, the remaining group is still huge. It's still bigger than a constant fraction of the total crowd.

Why This Matters

This result is a big deal because it answers a question that had been open for a long time. It tells us that the universe of unit fractions is more flexible than we thought. You can build a massive, structured set that avoids this specific arithmetic trap.

Furthermore, this discovery has a side effect. If you have a set of unit fractions where no two have an average that is a unit fraction, you automatically have a set with no "three-term arithmetic progressions" (no three numbers equally spaced). This improves upon previous records for how large such a set can be.

The author is careful to note that while this construction works and proves the set can be large, it might not be the largest possible set. There could be an even better, more complicated recipe out there waiting to be found. But for now, this proof is the definitive answer to the question: No, the set doesn't have to be small. It can be as big as a significant slice of the entire number line. The "no-average" club is open for business, and the membership is surprisingly large.

Technical Summary: Sets of Unit Fractions Without Two Members Whose Average is a Unit Fraction

Problem Statement
The paper addresses a question posed by Erdős and Graham regarding the maximum size of a subset $A \subseteq \{1, \dots, N\}$ satisfying a specific divisibility condition. The condition is that for any distinct $a, b \in A$ , the sum $a+b$ does not divide $2ab$. In the context of unit fractions, this is equivalent to stating that the average of any two distinct unit fractions $1/a$ and $1/b$ is not itself a unit fraction. Erdős and Graham asked whether such a set $A$ must necessarily have size $o(N)$ (i.e., whether the density of such sets must vanish as $N \to \infty$ ).

Additionally, the paper notes that this problem is closely related to the existence of large sets of unit fractions without non-trivial three-term arithmetic progressions, a question recently raised by Korsky.

Methodology and Construction
The author constructs a subset $A_N \subseteq \{1, \dots, N\}$ that violates the $o(N)$ conjecture, thereby providing a negative answer to the Erdős-Graham question. The construction relies on the function $\Omega(n)$ , which counts the total number of prime factors of $n$ with multiplicity.

The set $A_N$ is defined as the set of integers $a \in \{1, \dots, N\}$ such that for any $b \in \{1, \dots, N\}$ with $a \neq b$ and $\Omega(b) \le \Omega(a)$ , the condition $a+b \nmid 2ab$ holds.

The proof strategy involves the following steps:

Reduction to Divisibility: Using a change of variables $u = a/\gcd(a,b)$ and $v = b/\gcd(a,b)$ , the condition $a+b \mid 2ab$ is shown to be equivalent to the existence of coprime integers $u, v$ such that $u(u+v) \mid 2a$ and $b = av/u$.
Restriction to a "Good" Set $S$ : To ensure a large subset exists, the author restricts attention to a set $S$ $S$ of integers $a$ $a$ that satisfy three conditions:
- $a$ has no small prime factors (specifically, no prime factors less than a large parameter $L$ ).
- $a$ is not divisible by 4.
- The number of prime factors of $a$ up to $x$ , denoted $\Omega(a, x)$ , does not significantly exceed the expected average $(1+\epsilon)\log \log x$ for any $x$ .
Counting and Averaging: The core of the argument is to show that the "bad" elements in $S$ $S$ (those $a \in S$ $a \in S$ for which there exists a $b$ $b$ with $\Omega(b) \le \Omega(a)$ $Ω (b) \leq Ω (a)$ such that $a+b \mid 2ab$ $a + b ∣ 2 ab$ ) form a small proportion of $S$ $S$ .
- The author bounds the number of pairs $(u, v)$ such that $u(u+v)$ divides $2a$ for $a \in S$ .
- This involves estimating sums of multiplicative functions over dyadic intervals. The author utilizes a result by de la Bretèche and Tenenbaum regarding sums of non-negative multiplicative functions to bound the average number of such pairs.
Key Analytic Insight: The condition $\Omega(b) \le \Omega(a)$ (which translates to $\Omega(v) \le \Omega(u)$ in the change of variables) is crucial. Without this restriction, the average number of pairs would grow like a power of $\log \log N$ (similar to Hooley's $\Delta$ function). However, with this restriction and the exclusion of small prime factors, the average number of pairs for a given $a$ is bounded, allowing the "bad" set to be shown as a small fraction of $S$ .

Key Results

Theorem 1: There exists a constant $c > 0$ such that for all sufficiently large $N$ , the set $A_N$ defined above has size $|A_N| > cN$ .
Property of $A_N$ : For any distinct $a, b \in A_N$ , $a+b \nmid 2ab$ .
Implication for Unit Fractions: The set $\{1/a : a \in A_N\}$ is a set of unit fractions of linear density ($> cN$) such that no two distinct members have an average that is a unit fraction.
Implication for Arithmetic Progressions: Consequently, this set contains no non-trivial three-term arithmetic progressions of unit fractions.

Significance and Claims
The paper explicitly claims to provide a negative answer to the question of Erdős and Graham, demonstrating that sets of unit fractions with the specified property can have positive lower density, contrary to the suggestion that they must be $o(N)$ .

The author notes that while the construction provides a linear lower bound, the constant $c$ is not optimized. The paper suggests that the optimal lower bound likely arises from a more complex condition than the one used here. The work also improves upon a construction by Korsky regarding three-term arithmetic progressions in unit fractions.

The author acknowledges that the proof strategy was inspired by a calculation of Stijn Cambie for small $N$ and utilized AI tools (ChatGPT) for pattern recognition, reference searching, and proofreading, while the rigorous mathematical arguments and the specific adaptation of the strategy to the $\Omega(b) \le \Omega(a)$ condition are the author's contribution. The method also yields a lower bound for the related problem of sets where $a+b \nmid ab$ (sums are not unit fractions), though this bound is weaker than the trivial bound provided by the set of odd numbers.

The Great "No-Average" Heist

How the Magic Trick Works

Why This Matters

Technical Summary: Sets of Unit Fractions Without Two Members Whose Average is a Unit Fraction

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