On polynomial configurations in fractal sets

Abstract

We show that subsets of ${ℝ}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form

$$\left(x,x+A_{1}y,\dots ,x+A_{k-1}y,x+A_{k}y+Q\left(y\right)e_n\right),x\in {ℝ}^n,y\in {ℝ}^m,$$

where $A_i$ are real $n\times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n=\left(0,\dots ,0,1\right)$.