Remarks on dimension of unions of curves

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.