Two-point polynomial patterns in subsets of positive density in $\mathbb{R}^n$

TL;DR

This paper proves the existence of two-point polynomial configurations in positive density subsets of Euclidean space, providing explicit gap estimates and extending previous results on polynomial patterns and Roth-type theorems.

Contribution

It establishes new results on polynomial configurations in Euclidean spaces with explicit gap bounds, extending prior work on polynomial patterns and Roth theorems.

Findings

Existence of polynomial configurations with explicit gap estimates.

Extension of corner-type Roth theorem.

Decay estimates of oscillatory integral operators.

Abstract

Let $\gamma(t)=(P_1(t),\ldots,P_n(t))$ where $P_i$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+\gamma(t)\}$ in sets of positive density $\epsilon$ in $[0,1]^n$ with a gap estimate $t\geq \delta(\epsilon)$ when $P_i$'s are arbitrary, and in $[0,N]^n$ with a gap estimate $t\geq \delta(\epsilon)N^n$ when $P_i$'s are of distinct degrees where $\delta(\epsilon)=\exp\left(-\exp\left(c\epsilon^{-4}\right)\right)$ and $c$ only depends on $\gamma$. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain's reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.