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Large Sets of Integers with No Harmonic Triples

This paper establishes a new lower bound for the maximum size of a subset of $\{1, \dots, N\}$ that contains no distinct harmonic triples, achieved by constructing such a set via a random affine image of a dense progression-free set in a prime field followed by the removal of collapsed triples.

Original authors: Samuel Korsky

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Korsky

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 have a giant jar filled with numbered marbles, from 1 up to a very large number $N$ . Your goal is to pick out as many marbles as possible to keep in a smaller box, but with one strict rule: You cannot pick three marbles that form a "Harmonic Triple."

What is a Harmonic Triple?

In the world of numbers, a "Harmonic Triple" is a special trio of numbers $(a, b, c)$ where the reciprocals (the numbers flipped upside down, like $1/a$ ) form a perfect, evenly spaced line.

Think of it like a musical chord. If you have three notes, and the "distance" between the first and second note is exactly the same as the distance between the second and third, they are in harmony.

Mathematically, this means: $\frac{2}{a} = \frac{1}{b} + \frac{1}{c}$ .
If you find three numbers in your box that fit this equation, you have to throw the whole trio out.

The author, Samuel Korsky, asks: How big can our box get before we are forced to throw out so many numbers that it becomes tiny?

The Big Discovery

For a long time, mathematicians knew how to build large boxes of numbers that avoid standard patterns (like three numbers in a row: 3, 5, 7). But avoiding these "Harmonic" patterns was much harder because the math works differently (it's about flipping numbers, not just adding them).

Korsky proves that we can actually build a very large box of numbers that contains no Harmonic Triples.

The Size: The box isn't just a tiny speck; it's a significant chunk of the original jar.
The Catch: It's not quite as big as the standard "no-pattern" boxes, but it's still massive. The paper gives a specific formula showing that as the jar gets bigger, the box we can keep grows in a predictable, impressive way.

How Did He Do It? (The Construction)

Korsky didn't just pick numbers randomly. He used a clever, two-step "filtering" process, like a high-tech sieve.

Step 1: The "Shadow" Filter (The Prime Field)
Imagine you have a small, secret codebook (a small group of numbers called a "prime field"). In this codebook, there is a special list of numbers that already has no "three-in-a-row" patterns.
Korsky takes this small, perfect list and uses a random "magic lens" (a random affine image) to project it onto the giant jar of marbles.

He only keeps the marbles whose "shadows" (when you divide them by a specific prime number) land on that special list.
Because the original list had no patterns, most of the time, the marbles you pick won't form Harmonic Triples either.

Step 2: The "Collapse" Cleanup
Sometimes, the magic lens causes a glitch. A few bad triplets might slip through because they look different in the real world but look identical when viewed through the lens (they "collapse" into the same shadow).

Korsky realized these "glitch" triplets are rare. They only happen if the numbers share a very specific, rare mathematical property.
He calculated exactly how many of these glitches could exist. The number is small enough that he can simply throw away the "largest" number from every glitchy trio he finds.
Even after throwing these few away, the box is still huge.

The Analogy: The Party Guest List

Imagine you are hosting a party for $N$ guests. You want to invite as many people as possible, but you have a rule: No three guests can be "Harmonic."

The Standard Rule: Usually, you'd just avoid people who are standing in a straight line (like 1, 2, 3).
The Harmonic Rule: This is trickier. It's like saying, "No three guests can be related in a way that their 'inverse personalities' balance perfectly."
The Strategy:
- You first look at a small, VIP club (the prime field) where you know exactly who fits together.
- You use a random "name tag generator" to assign guests to the party based on who is in that VIP club.
- Most of the time, this works perfectly.
- Occasionally, three guests who shouldn't be together accidentally get the same name tag. You spot these few troublemakers and politely ask the tallest one to leave.
- Result: You still have a massive party, and no one is breaking the Harmonic rule.

What's Next?

The paper solves the "How big can we make it?" question (the lower bound). However, the author leaves one door open: Is it possible to have a party where almost everyone is invited (positive density)?
Currently, we don't know if it's possible to fill the box to the brim without breaking the rule, or if we are always forced to leave some people out. That remains a mystery for future mathematicians.

In short: We now know we can build a very large collection of numbers that avoids these tricky "Harmonic" relationships, using a smart mix of random selection and careful cleanup.

Technical Summary: Large Sets of Integers with No Harmonic Triples

Problem Statement
This paper investigates the maximum size, denoted $f(N)$ , of a subset $A \subseteq [N] = \{1, \dots, N\}$ that contains no distinct elements $a, b, c$ satisfying the harmonic triple condition:
$\frac{2}{a} = \frac{1}{b} + \frac{1}{c}$
This problem is presented as a reciprocal analogue to the classical problem of estimating $r_3(N)$ , the maximum size of a subset of $[N]$ containing no nontrivial three-term arithmetic progression (3-AP). While the upper and lower bounds for $r_3(N)$ have been extensively studied (referencing works by Behrend, Roth, Salem, Spencer, Elkin, Green, Wolf, Bloom, Sisask, Kelley, Meka, and others), the reciprocal map $x \mapsto 1/x$ is not an additive map on intervals of integers. Consequently, existing upper bounds for $r_3(N)$ do not directly apply to the harmonic setting.

Methodology and Construction
The author constructs a large subset of $[N]$ free of harmonic triples using a probabilistic method combined with a specific parametrization of harmonic triples. The construction proceeds as follows:

Prime Field Selection: A prime $q$ is chosen such that $q \asymp \log N$ .
Progression-Free Set: A dense three-term-progression-free set $R \subseteq \mathbb{F}_q$ is selected. The density of this set relies on the improved Behrend-type lower bound for $r_3(X)$ established by Elsholtz, Hunter, Proske, and Sauermann. Specifically, $|R|/q \ge \exp\left(-(C_{AP} + o(1))\sqrt{\log q}\right)$ , where $C_{AP} = 2\sqrt{\log(24/7)}$ .
Random Affine Filtering: A random affine image $R_{\lambda, \mu} = \lambda R + \mu$ (where $\lambda \in \mathbb{F}_q^\times, \mu \in \mathbb{F}_q$ ) is used to filter the odd integers $n \le N$ such that $n \not\equiv 0 \pmod q$ and $n^{-1} \pmod q \in R_{\lambda, \mu}$ . This initial random set $B_0$ has an expected size $\gg \delta N$ , where $\delta = |R|/q$ .
Handling Collapsed Triples: The core difficulty is that a harmonic triple in the integers might map to a constant progression (all residues equal) in $\mathbb{F}_q$ rather than a non-trivial progression. The paper utilizes a parametrization of odd harmonic triples (Lemma 2.2) to show that if a triple $b < a < c$ satisfies the harmonic condition and $a \equiv b \equiv c \pmod q$ , then the difference parameter $d$ in the parametrization must be divisible by $q$ .
Deletion Strategy: The construction deletes the largest element of any surviving harmonic triple. The number of such "collapsed" triples (where $q \mid d$ ) is shown to be $O(N \log N / q)$ . Since the probability of any specific triple surviving the random filter is $\delta$ , the expected number of deletions is small relative to the initial set size, provided the constant in $q \asymp \log N$ is chosen sufficiently large.

Key Results
The main theorem establishes a lower bound for $f(N)$ that mirrors the improved Behrend-type bound for $r_3(N)$ , but with a logarithmic shift:

Theorem 1.1: As $N \to \infty$ ,
$f(N) \gg N \exp\left( -(C_{AP} + o(1))\sqrt{\log \log N} \right)$
where $C_{AP} = 2\sqrt{\log(24/7)}$ .

The proof demonstrates that by choosing $q \asymp \log N$ , the term $\sqrt{\log q}$ in the exponent of the progression-free set density translates to $\sqrt{\log \log N}$ in the final bound for the harmonic set.

Significance and Open Questions
The paper claims to successfully transfer the state-of-the-art lower bounds for progression-free sets to the harmonic-triple-free setting. The construction is described as "direct," relying on the interplay between the density of progression-free sets in prime fields and the arithmetic structure of harmonic triples modulo a prime.

The author notes that the complementary upper-bound problem remains open. Specifically, the paper poses Question 1.2: Is it true that $f(N) = o(N)$ as $N \to \infty$ ? In other words, is it impossible for a set of positive density to be free of harmonic triples? The paper does not resolve this question but highlights it as the natural next step in the study of this problem.