Some comments on Laakso graphs and sets of differences

Abstract

We recall a variation of a construction due to Laakso \cite{LA}, also used by Lang and Plaut \cite{LA} of a doubling metric space $X$ that cannot be embedded into any Hilbert space. We give a more concrete version of this construction and motivated by the results of Olson \& Robinson \cite{OR}, we consider the Kuratowski embedding $\Phi (X)$ of $X$ into $L^{\infty }(X)$ and prove that $\Phi (X)-\Phi (X)$ is not doubling.