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

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.