On polynomial configurations in fractal sets

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.