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arXiv · 2208.03272

Remarks on dimension of unions of curves

Seheon Ham·Hyerim Ko·Sanghyuk Lee·Sewook Oh
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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 $\gamma$ 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.

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Seheon Ham, Hyerim Ko, Sanghyuk Lee, Sewook Oh. 2022-08-05. Remarks on dimension of unions of curves. https://arxiv.org/abs/2208.03272

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