论文标题
海森堡小组通过电影和点板事件的垂直投影
Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences
论文作者
论文摘要
令$ \ {π_{我们证明，如果$ k \ subset \ mathbb {h} $是borel设置的，则使用hausdorff dimension $ \ dim _ {\ mathbb {h}} k \ in [0,2] \ cup \ {3 \ {3 \} \ dim _ {\ mathbb {h}} k $$对于$ \ mathcal {h}^{1} $几乎每个$ e \ in s^{1} $。如果$ \ dim _ {\ mathbb {h}} k \ in [0,1] $，则此前早些时候知道这一点。 $ \ dim _ {\ mathbb {h}} k \ in [0,2] $和$ \ dim _ {\ mathbb {h}} k = 3 $的证明基于不同的技术。对于$ \ dim _ {\ mathbb {h}} k \在[0,2] $中，我们将物质减少到欧几里得问题，并应用了由于pramanik，Yang和Zahl而引起的电影功能的方法。 要处理情况，要处理$ \ dim _ {\ mathbb {h}} k = 3 $，我们在$ \ mathbb {r}^{3} $中引入了水平线和圆锥线之间的点线双重性。这使我们可以将海森堡问题转换为$ \ mathbb {r}^{3} $中的点板发病率问题。为了解决后者，由于Guth，Wang和Zhang，我们在$ \ mathbb {r}^{3} $中应用Kakeya不等式。此方法还为Borel设置$ K \ subset \ Mathbb {h} $带有$ \ dim _ {\ Mathbb {h}} k \ in（5/2,3）$的部分结果。
Let $\{π_{e} \colon \mathbb{H} \to \mathbb{W}_{e} : e \in S^{1}\}$ be the family of vertical projections in the first Heisenberg group $\mathbb{H}$. We prove that if $K \subset \mathbb{H}$ is a Borel set with Hausdorff dimension $\dim_{\mathbb{H}} K \in [0,2] \cup \{3\}$, then $$ \dim_{\mathbb{H}} π_{e}(K) \geq \dim_{\mathbb{H}} K $$ for $\mathcal{H}^{1}$ almost every $e \in S^{1}$. This was known earlier if $\dim_{\mathbb{H}} K \in [0,1]$. The proofs for $\dim_{\mathbb{H}} K \in [0,2]$ and $\dim_{\mathbb{H}} K = 3$ are based on different techniques. For $\dim_{\mathbb{H}} K \in [0,2]$, we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl. To handle the case $\dim_{\mathbb{H}} K = 3$, we introduce a point-line duality between horizontal lines and conical lines in $\mathbb{R}^{3}$. This allows us to transform the Heisenberg problem into a point-plate incidence question in $\mathbb{R}^{3}$. To solve the latter, we apply a Kakeya inequality for plates in $\mathbb{R}^{3}$, due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets $K \subset \mathbb{H}$ with $\dim_{\mathbb{H}} K \in (5/2,3)$.
