A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension

TL;DR

This paper proves a non-linear Roth theorem for fractal sets with large Hausdorff dimension, showing the existence of specific polynomial-configured point triples within such sets.

Contribution

It establishes a new non-linear combinatorial result for fractals of large dimension, extending Roth-type theorems to polynomial configurations.

Findings

Existence of polynomial-configured triples in large-dimensional fractals

Results hold for sets with Hausdorff dimension close to 1

Applicable to polynomials of degree d with no constant term

Abstract

Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - \epsilon_d < \dim_H(A) < 1$. Then for any polynomial $P$ of degree $d$ with no constant term, there exists a point configuration $\{ x, x-t,x-P(t) \} \subset A$ with $t \approx_P 1$.