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

Abstract

Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - ε_d < \dim_H(A) < 1$. 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 $t \approx_P 1$.