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

Abstract

Let ${\hat{I}}^3(t)=(P_1(t),\dots ,P_n(t))$ where $P_i$ is a real polynomial with zero constant term for each $1\le i\le n$. We will show the existence of the configuration $\{x,x+{\hat{I}}^3(t)\}$ in sets of positive density $\hat{I}\mu$ in $[0,1{]}^n$ with a gap estimate $t\ge \hat{I}´(\hat{I}\mu )$ when $P_i$'s are arbitrary, and in $[0,N{]}^n$ with a gap estimate $t\ge \hat{I}´(\hat{I}\mu )N^n$ when $P_i$'s are of distinct degrees where $\hat{I}´(\hat{I}\mu )=\exp \left(-\exp \left(c\hat{I}{\mu }^{-4}\right)\right)$ and $c$ only depends on ${\hat{I}}^3$. 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.

18 pages; we added the definition of the operator $T_{s,l}$ appearing in Lemma 2.2