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Three term rational function progressions in finite fields

Guo-Dong Hong, Zi Li Lim

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.

Three term rational function progressions in finite fields

Abstract

Let be rational functions such that and the constant function are linearly independent over , we prove an asymptotic formula for the number of the three term rational function progressions of the form in subsets of . 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.
Paper Structure (10 sections, 7 theorems, 74 equations)

This paper contains 10 sections, 7 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Introduction
  3. Acknowledgments
  4. Convention
  5. Notation
  6. Fourier Analysis Notation
  7. Algebraic Geometry Terminology
  8. Algebraic Geometry PET Induction
  9. Dimension Estimates
  10. Final Arguments

Key Result

Theorem 1.1

Let $F(t), G(t)\in \mathbb{Q}(t)$ be rational functions over $\mathbb{Q}$ such that $F(t), G(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$. Then, we have the asymptotic formula for all functions $f_0,f_1,f_2:\mathbb{F}_p\longrightarrow \mathbb{C}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Proposition 5.1
  • ...and 6 more