# Some comments on Laakso graphs and sets of differences

## TL;DR

The paper discusses a variation of Laakso's construction of a doubling metric space that cannot be embedded into Hilbert space, and shows that its Kuratowski embedding difference set is not doubling.

## Contribution

It provides a concrete version of Laakso's construction and analyzes the non-doubling property of the difference set of its Kuratowski embedding.

## Key findings

- Constructed a concrete Laakso-type space that cannot embed into Hilbert space.
- Proved that the difference set of the Kuratowski embedding of this space is not doubling.
- Highlights limitations of embedding doubling metric spaces into Hilbert spaces.

## 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.