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

Suppose that $d\ge 2$, 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$t \approx_P 1$.