  For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric
mapping $f\colon [0,1]^n \rightarrow X$ with the property that $f^{-1}$ is not
absolutely continuous with respect to the Hausdorff $n$-measure on $X$. That
is, there exists a Borel set $E \subset [0,1]^n$ with Lebesgue measure $|E|>0$
such that $f(E)$ has Hausdorff $n$-measure zero. The construction may be
carried out so that $X$ has finite Hausdorff $n$-measure and $|E|$ is
arbitrarily close to 1, or so that $|E| = 1$. This gives a negative answer to a
question of Heinonen and Semmes.
