论文标题
$ \ mathbb r^d $的薄亚集中点产品树
Trees of Dot Products in Thin Subsets of $\mathbb R^d$
论文作者
论文摘要
A. iosevich和K. Taylor表明，$ \ Mathbb r^d $的紧凑子集与Hausdorff尺寸大于$（D+1）/2 $的尺寸大于$（d+1）/2 $，在开放时间间隔内包含有间隙的树。在相同的维阈值下，我们证明了距离被点产物代替的类似结果。我们还表明，点产品的嵌入树的间隙在一组积极的Lebesgue度量中很普遍，对于Ahlfors-David常规集，具有给定间隙的树的数量与常规值定理一致。
A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalent in a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of trees with given gaps agrees with the regular value theorem.
