On polynomial configurations in fractal sets

TL;DR

This paper demonstrates that fractal sets with sufficiently large Hausdorff and Fourier dimensions necessarily contain complex polynomial configurations, extending understanding of pattern presence in fractal geometry.

Contribution

It establishes the existence of polynomial patterns in fractal sets based on their Hausdorff and Fourier dimensions, generalizing previous results to more complex polynomial configurations.

Findings

Fractal sets with large Hausdorff dimension contain polynomial configurations.

Sets with large Fourier dimension also contain these polynomial patterns.

The results apply to a broad class of polynomial patterns in Euclidean spaces.

Abstract

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 $e_n = (0,\dots,0,1)$.