  We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product
structure of hyperbolic sets} (2004), and by the third author and K. Simon in
\emph{Hausdorff and packing measures for solenoids} (2003), for
$C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $\Lambda$ over the
circle embedded in $\mathbb{R}^3$ non-conformally attracting in the stable
discs $W^s$ direction, to nonlinear ones. Under an assumption of transversality
and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing
\emph{thinness}, assuming also there is an invariant $C^{1+\varepsilon}$ strong
stable foliation, we prove that Hausdorff dimension ${\rm HD}(\Lambda\cap W^s)$
is the same quantity $t_0$ for all $W^s$ and else ${\rm HD}(\Lambda)=t_0+1$. We
prove also that for the packing measure $0<\Pi_{t_0}(\Lambda \cap W^s)<\infty$
but for Hausdorff measure ${\rm HM}_{t_0}(\Lambda\cap W^s)=0$ for all $W^s$.
Also $0<\Pi_{1+t_0}(\Lambda) <\infty$ and ${\rm HM}_{1+t_0}(\Lambda)=0$. A
technical part says that the holonomy along unstable foliation is locally
Lipschitz, except for a set of unstable leaves whose intersection with every
$W^s$ has measure ${\rm HM}_{t_0}$ equal to 0 and even Hausdorff dimension less
than $t_0$. The latter holds due to a large deviations phenomenon.
