Corners with polynomial side length
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912009760538624 |
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| author | Kravitz, Noah Kuca, Borys Leng, James |
| author_facet | Kravitz, Noah Kuca, Borys Leng, James |
| contents | A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In the case where $P$ has an integer root of multiplicity $1$, we show that if $A \subseteq [N]^2$ does not contain any nontrivial $P$-polynomial corners, then $$|A| \ll_P \frac{N^2}{(\log\log\log N)^c}$$ for some absolute constant $c>0$. This simultaneously generalizes a result of Shkredov about corner-free sets and a recent result of Peluse, Sah, and Sawhney about sets without $3$-term arithmetic progressions of common difference $z^2-1$. The main ingredients in our proof are a multidimensional quantitative concatenation result from our companion paper arXiv:2407.08636 and a novel degree-lowering argument for box norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Corners with polynomial side length Kravitz, Noah Kuca, Borys Leng, James Combinatorics Number Theory A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In the case where $P$ has an integer root of multiplicity $1$, we show that if $A \subseteq [N]^2$ does not contain any nontrivial $P$-polynomial corners, then $$|A| \ll_P \frac{N^2}{(\log\log\log N)^c}$$ for some absolute constant $c>0$. This simultaneously generalizes a result of Shkredov about corner-free sets and a recent result of Peluse, Sah, and Sawhney about sets without $3$-term arithmetic progressions of common difference $z^2-1$. The main ingredients in our proof are a multidimensional quantitative concatenation result from our companion paper arXiv:2407.08636 and a novel degree-lowering argument for box norms. |
| title | Corners with polynomial side length |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2407.08637 |