Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914837971337216 |
|---|---|
| author | Elsholtz, Christian Hunter, Zach Proske, Laura Sauermann, Lisa |
| author_facet | Elsholtz, Christian Hunter, Zach Proske, Laura Sauermann, Lisa |
| contents | We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of $\mathbb{F}_p^n$ for a fixed prime $p$ and large $n$, we prove a lower bound of $(cp)^n$ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields Elsholtz, Christian Hunter, Zach Proske, Laura Sauermann, Lisa Number Theory Combinatorics 11B25 We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of $\mathbb{F}_p^n$ for a fixed prime $p$ and large $n$, we prove a lower bound of $(cp)^n$ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem). |
| title | Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields |
| topic | Number Theory Combinatorics 11B25 |
| url | https://arxiv.org/abs/2406.12290 |