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Three term rational function progressions in finite fields

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arxiv 2401.01137 v1 pith:NRYSDGNX submitted 2024-01-02 math.NT math.AGmath.CAmath.CO

Guo-Dong Hong , Zi Li Lim This is my paper · ORCID
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abstract

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.

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  1. Uniform nonlinear Szemer\'{e}di theorem for corners in finite fields

    math.NT 2025-01 conditional novelty 7.0 of 10

    Corners generated by two independent rational functions in F_p^2 have the expected asymptotic count with uniform power saving p^{-1/40960}.

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