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Additive Ramsey theory over Piatetski-Shapiro numbers

This paper characterizes the partition regularity of linear equations over Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for specific ranges of $c$ depending on the number of variables, establishes quantitative density results, and strengthens Green's Fourier-analytic transference principle as adapted by Browning and Prendiville.

Original authors: Jonathan Chapman, Sam Chow, Philippa Holdridge

Published 2026-05-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jonathan Chapman, Sam Chow, Philippa Holdridge

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 bag of marbles, but these aren't just any marbles. They are special "Piatetski-Shapiro" marbles. To make one, you take a number $n$ , raise it to a specific power $c$ (where $c$ is a number between 1 and 2), and then round down to the nearest whole number.

If $c=2$ , you get perfect squares (1, 4, 9, 16...). If $c$ is something like 1.1, you get a different, slightly more scattered set of numbers.

The authors of this paper are playing a game of "coloring" with these marbles.

The Big Game: The Color-Blind Equation

Imagine you have a huge pile of these special marbles. You decide to paint them using a finite number of colors (say, Red, Blue, and Green). You paint them randomly, or perhaps in a very tricky pattern, but you use only a few colors.

The question is: No matter how you paint them, will you always be forced to create a "monochromatic" solution to a math equation?

A "monochromatic solution" means you find three (or more) marbles that are all the same color and satisfy a simple equation like $x + y = z$ (a "Schur triple").

The Old Rule: For normal whole numbers, mathematicians have known since 1916 that if you color them with enough colors, you always find a red $x$ , red $y$ , and red $z$ where $x+y=z$ .
The New Challenge: Does this still hold true for our special "Piatetski-Shapiro" marbles?

The Discovery: It Depends on the "Power"

The authors, Chapman, Chow, and Holdridge, figured out exactly when this rule works for these special marbles.

They found that it depends on the "power" $c$ you used to create the marbles and how many variables are in your equation (how many marbles you need to add up).

The Sweet Spot: If the power $c$ is small enough (specifically, if it's less than a certain threshold like $12/11$ for three-marble equations), then YES. No matter how you paint the marbles, you cannot avoid finding a group of the same color that solves the equation.
The Limit: If the power $c$ gets too close to 2 (getting too close to perfect squares), the math gets too messy, and they can't guarantee the rule holds yet.

Think of it like a game of musical chairs. If the chairs (the numbers) are spaced out just right (low $c$ ), no matter how you shuffle the players (the colors), someone will always end up sitting with their friends in the same group. If the chairs are spaced too differently (high $c$ ), the players might manage to avoid each other.

The "Density" Puzzle: How Empty Can a Room Be?

The paper also tackles a related question about "density."

Imagine you have a room full of these special marbles. You want to pick out a group of marbles to put in a smaller box, but you have a rule: You are not allowed to pick any group that solves the equation $x + y = z$ . (You want a "solution-free" group).

The Question: How big can this "forbidden" box get? Can you fill half the room? A quarter?
The Answer: The authors prove that if you try to avoid these solutions, your box must be incredibly empty compared to the whole room. As the room gets bigger, the "forbidden" box becomes a tiny, tiny fraction of the total.

They didn't just say "it's small"; they gave a very precise mathematical formula for how small it gets. They found that their method allows them to prove the box is much emptier than previous mathematicians could prove for similar problems involving perfect squares.

How Did They Do It? (The Magic Tools)

To solve this, the authors used a sophisticated toolkit from a field called "Additive Combinatorics." You can think of their method as a three-step magic trick:

The Weighted Lens: Instead of looking at the marbles directly, they put on a special pair of glasses (a "weight function") that makes the special marbles look more like normal numbers. This helps them use tools designed for normal numbers.
The Frequency Filter: They analyzed the marbles using "Fourier analysis" (which is like looking at the marbles through a prism to see their hidden frequencies). They proved that these special marbles have a very specific, smooth pattern in their frequencies, which makes them predictable.
The Transfer Principle: This is the most important trick. They took a powerful theorem that works for normal numbers (Green's Transference Principle) and updated it. They showed that because their "weighted glasses" are so good, if a rule works for normal numbers, it must also work for these special marbles, provided the "power" $c$ isn't too high.

Summary

In simple terms, this paper is a victory for order over chaos. It proves that even in a set of numbers that looks a bit weird and scattered (the Piatetski-Shapiro numbers), the fundamental laws of arithmetic (like $x+y=z$ ) are so strong that you cannot hide them, no matter how you try to color the numbers or pick a subset to avoid them.

They also improved the mathematical "magnifying glass" used to see these patterns, making the proof sharper and the results stronger than ever before for these specific types of numbers.

Technical Summary: Additive Ramsey Theory over Piatetski-Shapiro Numbers

Problem Statement
The paper investigates partition regularity and density results for linear equations over the set of Piatetski-Shapiro numbers, defined as $\mathcal{P}\mathcal{S}_c = \{ \lfloor n^c \rfloor : n \in \mathbb{N} \}$ for a parameter $c > 1$ . Specifically, the authors address whether linear equations of the form $c_1 x_1 + \dots + c_s x_s = 0$ (where $\sum c_i = 0$ ) possess non-trivial monochromatic solutions when the set $\mathcal{P}\mathcal{S}_c$ is partitioned into finitely many color classes.

This work extends classical results:

Schur's Theorem (1916): Guarantees monochromatic solutions to $x+y=z$ in $\mathbb{N}$ .
Rado's Criterion (1933): Characterizes partition regularity for linear systems.
Problem A (Erdős–Graham): Asks for monochromatic Pythagorean triples ( $x^2+y^2=z^2$ ). While solved for $s \ge 5$ variables, the case for squares ( $c=2$ ) remains open for $s=3$ .
Problem B: The authors frame their work as an attempt to solve a version of Problem A for Piatetski-Shapiro numbers, asking if the conclusion of Corollary 1.5 (monochromatic Schur triples) holds for any $c \in [1, 2]$ .

Methodology
The authors employ the Fourier-analytic transference principle, pioneered by Green, adapted to the sparse setting of Piatetski-Shapiro numbers. The core strategy involves:

Weight Function Construction: Defining a weight function $\nu$ supported on $\mathcal{P}\mathcal{S}_c(N)$ such that $\|\nu\|_1 \sim N$ .
Fourier Decay: Establishing that the Fourier transform of $\nu$ approximates that of the indicator function of the interval $[N]$ , specifically bounding $\|\hat{\nu} - \hat{1}_{[N]}\|_\infty$ .
Restriction Estimates: Proving bounds on the $L^t$ norm of the Fourier transform of functions dominated by $\nu$ .
Transference: Comparing the weighted count of solutions in $\mathcal{P}\mathcal{S}_c$ with the unweighted count in $[N]$ .

A significant methodological contribution is the strengthening of the transference principle (specifically Browning and Prendiville's version of Green's principle). By incorporating recent advances in the quantitative density of sets free of arithmetic progressions (Kelley–Meka, Bloom–Sisask) and pseudorandomness (Ko'sciuszko, Filmus et al.), the authors improve the quantitative bounds of the transference machinery.

Key Contributions and Results

Partition Regularity (Theorem 1.4):
The authors characterize partition regularity for linear equations over $\mathcal{P}\mathcal{S}_c$ when $1 < c < c^\dagger(s)$ , where $s \ge 3$ is the number of variables. The threshold $c^\dagger(s)$ is defined as:
- $c^\dagger(3) = 12/11$
- $c^\dagger(4) = 7/6$
- $c^\dagger(s) = 2$ for $s \ge 5$ .
Under these conditions, any finite partition of $\mathcal{P}\mathcal{S}_c$ contains a monochromatic non-trivial solution to $c_1 x_1 + \dots + c_s x_s = 0$ . Notably, for $s=3$ , this resolves a version of the Schur triple problem for $c < 12/11$ .
Quantitative Density Bounds (Theorem 1.8):
The paper establishes that if a subset $A \subseteq \mathcal{P}\mathcal{S}_c(N)$ contains no non-trivial solutions to the linear equation, its density is bounded by:
$|A| \le \frac{|\mathcal{P}\mathcal{S}_c(N)|}{\exp(C^{-1}(\log \log N)^{1/\tau(s)})}$
where $\tau(3)=9$ and $\tau(s)=7$ for $s \ge 4$ .

This result improves upon previous work by Browning and Prendiville for squares ( $c=2$ ), which yielded a saving of $(\log \log \log N)^{O(1)}$ . The authors attribute this improvement to two factors:
- The elimination of the need for the "W-trick" in this specific setting.
- The strengthened Fourier-analytic transference principle.
Refined Transference Principle (Theorem 4.3):
The authors update the transference principle from [6, Proposition 2.8], incorporating the quantitative rates of pseudorandomness derived from recent breakthroughs in additive combinatorics. This allows for sharper density bounds in sparse sets.

Significance and Claims
The paper claims to solve a version of Problem A (monochromatic Pythagorean triples) for Piatetski-Shapiro numbers with parameter $c < 12/11$ (Corollary 1.5). It highlights that the threshold $12/11$ coincides with a threshold Piatetski-Shapiro himself found in proving a version of the prime number theorem for these numbers.

The authors emphasize that their density saving is "substantially greater" than that obtained for squares, primarily due to the improved transference principle and the specific analytic properties of Piatetski-Shapiro numbers which allow for more precise exponential sum estimates. While they note that Saito and Yoshida previously settled the density analogue for $c \in [1, 2)$ using Szemerédi's theorem, the authors assert that their result provides a superior quantitative density bound.

The work also briefly touches on Piatetski-Shapiro primes, noting that while Mirek established density results for 3AP-free sets in this context, the existence of monochromatic Schur triples of odd numbers (and thus primes) is impossible, though a shifted equation $x+y=z+1$ can be shown to have solutions.