A note on arithmetic progressions with restricted differences
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909040416653312 |
|---|---|
| author | Conlon, David Fox, Jacob Pham, Huy Tuan |
| author_facet | Conlon, David Fox, Jacob Pham, Huy Tuan |
| contents | In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd prime power, there is $\varepsilon_q>0$ such that if $S \subseteq \mathbb{F}_q$ with $0 \in S$ and $|S|>(q+1)/2$ and $A \subseteq \mathbb{F}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $|A| \leq q^{(1-\varepsilon_q)n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13628 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on arithmetic progressions with restricted differences Conlon, David Fox, Jacob Pham, Huy Tuan Combinatorics Number Theory In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd prime power, there is $\varepsilon_q>0$ such that if $S \subseteq \mathbb{F}_q$ with $0 \in S$ and $|S|>(q+1)/2$ and $A \subseteq \mathbb{F}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $|A| \leq q^{(1-\varepsilon_q)n}$. |
| title | A note on arithmetic progressions with restricted differences |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2605.13628 |