# Polynomial Roth theorems on sets of fractional dimensions

## TL;DR

This paper proves that certain fractal sets of fractional dimension in the real line contain polynomial configurations, extending previous results and answering open questions about polynomial Roth theorems in fractal sets.

## Contribution

It extends polynomial Roth theorems to fractal sets with specific measure and Fourier decay conditions, broadening the scope of polynomial configuration results.

## Key findings

- Sets of fractional Hausdorff dimension contain polynomial configurations.
- Supports for measures with Fourier decay imply existence of polynomial patterns.
- Generalizes previous results to polynomial settings in fractal sets.

## Abstract

Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $\alpha\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Laba and the third author to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot, Laba and the third author.