Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if \({f\colon X\to Y}\) is a quasisymmetric mapping between Ahlfors regular spaces, then \({dim_H f(E)\leq dim_H E}\) 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 \({\mathbb{R}^N}\), \({N\geq 2}\), 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 \({E \subset {\mathbb{R}}}\) is Ahlfors \({d}\)-regular, \({d < 1}\), then some component of \({f(E \times {\mathbb{R}})}\) has dimension at most \({2/(d+1)}\), 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 et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric, 2016).