Additive Ramsey theory over Piatetski-Shapiro numbers
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 91-110

We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers $\lfloor n^c \rfloor $ when $1 < c < c^\dagger (s)$, where $s \geqslant 3$ is the number of variables. Here $c^\dagger (3) = 12/11$ and $c^\dagger (4) = 7/6$, while $c^\dagger (s) = 2$ 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.

Nous caractérisons la régularité de partitions pour les équations linéaires en nombres de Piatetski-Shapiro $\lfloor n^c \rfloor $ lorsque $1 < c < c^\dagger (s)$, où $s \geqslant 3$ est le nombre de variables. Ici $c^\dagger (3) = 12/11$ et $c^\dagger (4) = 7/6$, tandis que $c^\dagger (s) = 2$ pour $s \geqslant 5$. Nous établissons également des résultats de densité avec des bornes quantitatives. Suite aux développements récents, nous saisissons cette occasion pour mettre à jour la version de Browning et Prendiville du principe de transfert analytique de Fourier de Green, en renforçant ainsi sa conclusion.

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DOI : 10.5802/jtnb.1356
Classification : 11B30, 05D10, 11D72, 11L15
Keywords: Arithmetic combinatorics, Ramsey theory, Diophantine equations, Hardy–Littlewood method, exponential sums, partition regularity, restriction theory
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Jonathan Chapman; Sam Chow; Philippa Holdridge. Additive Ramsey theory over Piatetski-Shapiro numbers. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 91-110. doi: 10.5802/jtnb.1356
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