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

Xuezhi Chen, Changxing Miao

Let $\gamma (t)=(P_1(t),\dots ,P_n(t))$ 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 $\{x,x+\gamma (t)\}$…