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 $\alpha\in (0, d-1)$.

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.

This talk is based on recent work with Herym Ko, Sanghyuk Lee, and Sewook Oh.