A proof of the Erdös similarity conjecture

Abstract

We show that for any infinite set $A$ in $\mathbb{R}$, there exists a compact set $E\subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. This proves the Erdös similarity conjecture.

A gap was found in Proposition 3.3 of the paper. We would like to thank professor Vjekoslav Kovac for pointing this out. Therefore, we would like to withdraw the paper