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arxiv: 2410.14427 · v4 · pith:73LZIFELnew · submitted 2024-10-18 · 🧮 math.NT · math.CO

Additive Ramsey theory over Piatetski-Shapiro numbers

Jonathan Chapman , Sam Chow , Philippa Holdridge This is my paper

Pith reviewed 2026-05-23 18:48 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords partition regularityPiatetski-Shapiro numbersadditive Ramsey theorylinear equationsFourier-analytic transferencedensity resultssparse sets
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The pith

Linear equations remain partition regular over Piatetski-Shapiro numbers floor(n^c) below explicit thresholds depending on the number of variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper characterises when linear equations are partition regular when all variables are restricted to the Piatetski-Shapiro numbers, which are the integers closest to n raised to a power c. The thresholds are c less than 12/11 for three variables, 7/6 for four variables, and 2 for five or more variables. A sympathetic reader would care because these sets become sparser as c increases, so the result shows how sparse a set can be while still guaranteeing monochromatic solutions to linear equations. The work also provides quantitative density estimates for the number of solutions and strengthens an existing tool from Fourier analysis for transferring results from dense sets to sparse ones.

Core claim

We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers floor(n^c) when 1 < c < c^dag(s), where s >= 3 is the number of variables. Here c^dag(3) = 12/11 and c^dag(4) = 7/6, while c^dag(s) = 2 for s >= 5. We also establish density results with quantitative bounds. Following recent developments, we update Browning and Prendiville's version of Green's Fourier-analytic transference principle, strengthening its conclusion.

What carries the argument

The strengthened Fourier-analytic transference principle, which allows transferring partition regularity from the integers to the Piatetski-Shapiro numbers.

Load-bearing premise

The strengthened transference principle can be applied in the Piatetski-Shapiro setting to obtain the partition regularity results.

What would settle it

A concrete linear equation in three variables with no solutions in the Piatetski-Shapiro numbers for some c slightly less than 12/11, while solutions exist in the full integers.

read the original abstract

We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ when $1 < c < c^\dag(s)$, where $s \geqslant 3$ is the number of variables. Here $c^\dag(3) = 12/11$ and $c^\dag(4) = 7/6$, while $c^\dag(s) = 2$ for $s \geqslant 5$. We also establish density results with quantitative bounds. Following recent developments, we take this opportunity to update Browning and Prendiville's version of Green's Fourier-analytic transference principle, strengthening its conclusion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript characterizes the partition regularity of linear equations in s variables over the Piatetski-Shapiro sequence {⌊n^c⌋} precisely when 1 < c < c^dag(s), with explicit values c^dag(3) = 12/11, c^dag(4) = 7/6 and c^dag(s) = 2 for s ≥ 5. It also supplies quantitative density results and strengthens the conclusion of Browning–Prendiville’s Fourier-analytic transference principle.

Significance. If the strengthened transference principle applies to the Piatetski-Shapiro sequence at the stated thresholds, the work supplies the first explicit characterization of this form for a sparse, non-polynomial set and furnishes a reusable strengthening of the transference tool. The quantitative density bounds are a further concrete contribution.

minor comments (2)
  1. The abstract states the thresholds without indicating how the major/minor-arc estimates are optimized to produce exactly 12/11 and 7/6; a one-sentence pointer to the relevant parameter choice in the proof would improve readability.
  2. Notation for the updated transference principle (e.g., the precise form of the pseudorandomness hypothesis) should be introduced with a displayed equation early in the paper rather than only in the body of the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The report accurately summarizes the main results on the range of c for which linear equations are partition regular over the Piatetski-Shapiro sequence, the explicit thresholds, the density bounds, and the strengthened transference principle.

Circularity Check

0 steps flagged

No circularity: thresholds derived from strengthened transference principle applied to sequence properties

full rationale

The paper updates Browning and Prendiville's Fourier-analytic transference principle (explicitly stated as an update in the abstract) and applies the strengthened version to obtain partition regularity for linear equations over floor(n^c) when c < c^dag(s), with the explicit values c^dag(3)=12/11 and c^dag(4)=7/6 arising from the analysis of major/minor arc estimates or pseudorandomness conditions for the Piatetski-Shapiro sequence. No quoted step reduces a claimed prediction or first-principles result to a fitted input, self-definition, or load-bearing self-citation chain by construction. The derivation is self-contained against the external benchmark of the prior transference principle, which is independent of the present thresholds.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claims rest on the applicability of the updated transference principle whose details are not visible.

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