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A bilinear estimate in $\mathbb{F}_p$

Necef Kavrut, Shukun Wu

Abstract

We improve an $L^2\times L^2\to L^2$ estimate for a certain bilinear operator in the finite field of size $p$, where $p$ is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a corollary, we show that there exists a quadratic progression $x,x+y,x+y^2$ for nonzero $y$ inside any subset of $\mathbb{F}_p$ of density $\gtrsim p^{-1/8}$

A bilinear estimate in $\mathbb{F}_p$

Abstract

We improve an estimate for a certain bilinear operator in the finite field of size , where is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a corollary, we show that there exists a quadratic progression for nonzero inside any subset of of density
Paper Structure (3 sections, 5 theorems, 65 equations)

This paper contains 3 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Ending remarks

Key Result

Theorem 1.2

Given $f_1,f_2: \mathbb{F}_p :\to \mathbb{C}$ one has

Theorems & Definitions (9)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Dong-Li-Sawin Theorem 3.1
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof