论文标题
海森伯格集团中正确的coset预测的维度扭曲
Dimension Distortion by Right Coset Projections in the Heisenberg Group
论文作者
论文摘要
我们研究了海森伯格集团（Heisenberg Group）中的纤维纤维的垂直投影家族，其纤维是水平飞机的合适套筒，$ \ mathbb {h}^n $。我们证明，相对于欧几里得公制，以及在这种情况下，我们显示的行为与欧几里得指标相似，在这些映射下集合的Hausdorff尺寸变形的下限。我们的边界在尺寸范围的很大一部分中都很清晰，我们为剩余范围提供了猜想的锋利下限。我们的方法还使我们能够改善已知的垂直投影属于$ \ mathbb {h}^n $的已知下限，以$ n \ geq 2 $。
We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric which we show behaves like the Euclidean metric in this context. Our bounds are sharp in a large part of the dimension range, and we give conjectural sharp lower bounds for the remaining range. Our approach also lets us improve the known almost sure lower bound for the standard family of vertical projections in $\mathbb{H}^n$ for $n \geq 2$.
