We extend the results of Hasselblatt and Schmeling [Dimension product structure of hyperbolic sets. Modern Dynamical Systems and Applications. Eds. B. Hasselblatt, M. Brin and Y. Pesin. Cambridge University Press, New York, 2004, pp. 331–345] and of Rams and Simon [Hausdorff and packing measure for solenoids. Ergod. Th. & Dynam. Sys. 23 (2003), 273–292] 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 solenoids. Under the assumptions of transversality and on the Lyapunov exponents for an appropriate Gibbs measure imposing thinness, as well as the assumption that there is an invariant
$C^{1+\varepsilon }$ strong stable foliation, we prove that Hausdorff dimension
$\operatorname {\mathrm {HD}}(\Lambda \cap W^s)$ is the same quantity
$t_0$ for all
$W^s$ and else
$\mathrm {HD}(\Lambda )=t_0+1$. We prove also that for the packing measure,
$0<\Pi _{t_0}(\Lambda \cap W^s)<\infty $, but for Hausdorff measure,
$\mathrm {HM}_{t_0}(\Lambda \cap W^s)=0$ for all
$W^s$. Also
$0<\Pi _{1+t_0}(\Lambda ) <\infty $ and
$\mathrm {HM}_{1+t_0}(\Lambda )=0$. 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
$\mathrm {HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than
$t_0$. The latter holds due to a large deviations phenomenon.