Quantitative concatenation for polynomial box norms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912836015357952 |
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| author | Kravitz, Noah Kuca, Borys Leng, James |
| author_facet | Kravitz, Noah Kuca, Borys Leng, James |
| contents | Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper arXiv:2407.08637, we complete this program for sets in $[N]^2$ lacking nondegenerate progressions of the form $(x, y), (x + P(z), y), (x, y + P(z))$, where $P \in \mathbb{Z}[z]$ is any fixed polynomial with an integer root of multiplicity $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_08636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative concatenation for polynomial box norms Kravitz, Noah Kuca, Borys Leng, James Combinatorics Number Theory Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper arXiv:2407.08637, we complete this program for sets in $[N]^2$ lacking nondegenerate progressions of the form $(x, y), (x + P(z), y), (x, y + P(z))$, where $P \in \mathbb{Z}[z]$ is any fixed polynomial with an integer root of multiplicity $1$. |
| title | Quantitative concatenation for polynomial box norms |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2407.08636 |