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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 $\\dim_H f(y+E) = \\dim_H E$. 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Bishop, Hrant Hakobyan, Marshall Williams","submitted_at":"2012-11-01T17:23:35Z","abstract_excerpt":"We show that if $f:X\\to Y$ 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 $\\dim_H f(y+E) = \\dim_H E$. 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