Two-Point Polynomial Patterns in Subsets of Positive Density in Rⁿ

Let γ(t)=(P₁(t),…,Pₙ(t)) where Pᵢ is a real polynomial with zero constant term for each 1≤ i≤ n. We will show the existence of the configuration ,x+γ(t)\ in sets of positive density ε in [0,1]ⁿ with a gap estimate t≥ δ(ε) when Pᵢ's are arbitrary, and in [0,N]ⁿ with a gap estimate t≥ δ(ε)Nⁿ when Pᵢ's are of distinct degrees where δ(ε)=exp(-exp(cε⁻⁴)) and c only depends on γ. 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.