Three term rational function progressions in finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909059669557248 |
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| author | Hong, Guo-Dong Lim, Zi Li |
| author_facet | Hong, Guo-Dong Lim, Zi Li |
| contents | Let $F(t),G(t)\in \mathbb{Q}(t)$ be rational functions such that $F(t),G(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the three term rational function progressions of the form $x,x+F(y),x+G(y)$ in subsets of $\mathbb{F}_p$. The main new ingredient is an algebraic geometry version of PET induction that bypasses Weyl's differencing. This answers a question of Bourgain and Chang. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three term rational function progressions in finite fields Hong, Guo-Dong Lim, Zi Li Number Theory Algebraic Geometry Classical Analysis and ODEs Combinatorics Let $F(t),G(t)\in \mathbb{Q}(t)$ be rational functions such that $F(t),G(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the three term rational function progressions of the form $x,x+F(y),x+G(y)$ in subsets of $\mathbb{F}_p$. The main new ingredient is an algebraic geometry version of PET induction that bypasses Weyl's differencing. This answers a question of Bourgain and Chang. |
| title | Three term rational function progressions in finite fields |
| topic | Number Theory Algebraic Geometry Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2401.01137 |