On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911869710630912 |
|---|---|
| author | Pliego, Javier |
| author_facet | Pliego, Javier |
| contents | When $g\in\mathbb{N}$ we say that $A\subset\mathbb{N}$ is a $B_{2}[g]$ sequence if every $m\in\mathbb{N}$ has at most $g$ distinct representations of the shape $m=b_{1}+b_{2}$ with $b_{1}\leq b_{2}$ and $b_{1},b_{2}\in A$. We show for every $0<\varepsilon<1$ that whenever $g>\frac{1}{\varepsilon}$ then there is a $B_{2}[g]$ sequence $A$ having the property that every sufficiently large $n\in\mathbb{N}$ can be written as $$n=a_{1}+a_{2}+a_{3},\ \ \ \ \ \ \ \ \ a_{3}\leq n^{\varepsilon}\ \ \ \ \ \ \ \ \ a_{i}\in A,$$ and satisfying for large $x$ the estimate $$\lvert A\cap [1,x]\rvert\gg x^{g/(2g+1)}.$$ The above lower bound improves upon earlier results of Cilleruelo and of Erdős and Renyi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences Pliego, Javier Number Theory Combinatorics When $g\in\mathbb{N}$ we say that $A\subset\mathbb{N}$ is a $B_{2}[g]$ sequence if every $m\in\mathbb{N}$ has at most $g$ distinct representations of the shape $m=b_{1}+b_{2}$ with $b_{1}\leq b_{2}$ and $b_{1},b_{2}\in A$. We show for every $0<\varepsilon<1$ that whenever $g>\frac{1}{\varepsilon}$ then there is a $B_{2}[g]$ sequence $A$ having the property that every sufficiently large $n\in\mathbb{N}$ can be written as $$n=a_{1}+a_{2}+a_{3},\ \ \ \ \ \ \ \ \ a_{3}\leq n^{\varepsilon}\ \ \ \ \ \ \ \ \ a_{i}\in A,$$ and satisfying for large $x$ the estimate $$\lvert A\cap [1,x]\rvert\gg x^{g/(2g+1)}.$$ The above lower bound improves upon earlier results of Cilleruelo and of Erdős and Renyi. |
| title | On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2405.04154 |