  We prove a formula relating the Hausdorff dimension of a deterministic Borel
subset of $\mathbb R$ and the Hausdorff dimension of its image under a
conformal map from the upper half-plane to a complementary connected component
of an SLE$_\kappa$ curve for $\kappa \not =4$. Our proof is based on the
relationship between SLE and Liouville quantum gravity together with the
one-dimensional KPZ formula of Rhodes-Vargas (2011) and the KPZ formula of
Gwynne-Holden-Miller (2015). As an intermediate step we prove a KPZ formula
which relates the Euclidean dimension of a subset of an SLE$_\kappa$ curve for
$\kappa \in (0,4)\cup(4,8)$ and the dimension of the same set with respect to
the $\gamma$-quantum natural parameterization of the curve induced by an
independent Gaussian free field, $\gamma = \sqrt \kappa \wedge
(4/\sqrt\kappa)$.
