Remarks on dimension of unions of curves

TL;DR

This paper investigates the Hausdorff dimension of unions of curves generated by translation and dilation, extending classical circle packing problems to higher dimensions and fractal measures.

Contribution

It establishes lower bounds on the Hausdorff dimension of unions of dilated and translated curves in higher dimensions, generalizing previous results to fractal and multi-parameter settings.

Findings

Unions of curves have Hausdorff dimension at least α+1 if the generating set has dimension greater than α.

Results extend to multi-parameter dilations of curves.

Key use of local smoothing estimates for averages over fractal measures.

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.