论文标题
关于曲线工会维度的评论
Remarks on dimension of unions of curves
论文作者
论文摘要
我们研究了Marstand的圆形包装问题的类似物，以示为较高维度的曲线。我们考虑曲线的集合，这些曲线是通过翻译和扩张的曲线$γ$在$ \ mathbb r^d $中，即$ x +tγ$，$（x，t）\ in \ mathbb r^d \ times（0，\ infty）$。对于Borel集合$ f \ subset \ mathbb r^d \ times（0，\ infty）$，我们显示曲线的工会$ \ bigCup _ {（x，x，x，t）\ in F}（x+tγ）$具有hausdorff dive hausdorff dimemension至少$α+f $ f $ f $ f $ hausdormension $α D-1）$。我们还获得了由$γ$的多参数扩张产生的曲线工会的结果。主要成分之一是相对于分形测量的局部平滑型估计（对于平均值）。
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 $\mathbb R^d$, i.e., $ x + t γ$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+tγ)$ has Hausdorff dimension at least $α+1$ whenever $F$ has Hausdorff dimension bigger than $α$, $α\in (0, d-1)$. 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.
