Abstract

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 γ in Rd, i.e., x+tγ, (x,t)∈Rd×(0,∞). For a Borel set F⊂Rd×(0,∞), we show the unions of curves ⋃(x,t)∈F(x+tγ) has Hausdorff dimension at least α+1 whenever F has Hausdorff dimension bigger than α, α∈(0,d−1). We also obtain results for unions of curves generated by multi-parameter dilation of γ. One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures.

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