Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if f: X→ Y is a quasisymmetric mapping between Ahlfors regular spaces, then dim _Hf(E) ≤ dim _HE for “almost every” bounded Ahlfors regular set E⊆ X. If additionally, X and Y are Loewner spaces then dim _Hf(E) = dim _HE for “almost every" Ahlfors regular set E⊂ 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 R^N, N≥ 2 , then for Lebesgue a.e. y∈ R^N we have dim _Hf(y+ E) = dim _HE. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E⊂ R is Ahlfors d-regular, d< 1 , then some component of f(E× 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⊆ R and planar quasiconformal map f such that f(R× 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).