  We study an analogue of Marstrand's circle packing problem for curves in
higher dimensions. We consider collections of curves which are generated by
translation and dilation of a curve $\gamma$ in $\mathbb R^d$, i.e., $ x + t
\gamma$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset
\mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in
F} ( x+t\gamma )$ has Hausdorff dimension at least $\alpha+1$ whenever $F$ has
Hausdorff dimension bigger than $\alpha$, $\alpha\in (0, d-1)$. We also obtain
results for unions of curves generated by multi-parameter dilation of $\gamma$.
One of the main ingredients is a local smoothing type estimate (for averages
over curves) relative to fractal measures.
