Qusisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then ${\dim }_{H}f(E)\le {\dim }_{H}E$ for “almost every” bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then ${\dim }_{H}f(E)={\dim }_{H}E$ for “almost every” Ahlfors regular set $E\subset X$. The precise statements of these results are given in terms of Fuglede’s modulus of measures. As a corollary of these general theorems we show that if $f$ is a quasiconformal map of ${\mathbb{R}}^N$, $N\ge 2$, then for Lebesgue a.e. $y\in {\mathbb{R}}^N$ we have ${\dim }_{H}f(y+E)={\dim }_{H}E$. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if $E \subset $\mathbb{R}$$isAhlfors$d$-regular,$d<1$,thensomecomponentof$f(E \times $\mathbb{R}$)$hasdimensionatmost$2/(d+1)$, and we construct examples to show this bound is sharp. In addition, we show there is a $1$-dimensional set $S\subseteq \mathbb{R}$ and planar quasiconformal map $f$ such that $f(\mathbb{R}\times S)$ contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson \cite{Tyson:frequency} and answer questions posed in \cite{Tyson:frequency} and \cite{AimPL}.