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

Abstract

Suppose that $d\ge 2$, 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 $t{\approx }_{P}1$.

Cited by in corpus (2)

• Polynomial progressions in topological fields
• Two bipolynomial Roth theorems in $\mathbb{R}$