论文标题
Bourgain投影定理的非线性版本
A nonlinear version of Bourgain's projection theorem
论文作者
论文摘要
我们证明了Bourgain的投影定理，用于$ c^2 $地图的参数化家族，即使在线性案例中也可以完善原始语句。 As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to $1$ in $\mathbb{R}^2$ or close to $3/2$ in $\mathbb{R}^3$, then for $y\in A$ outside of a very sparse set, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff尺寸至少$ 1/2+c $，其中$ c $是通用的。此外，如果相对于$ c^2 $正态曲率的规范，则相同。随着进一步的应用，我们在球形投影的尺寸上获得了新的界限，并在相当普遍的假设下，对$δ$ - 鲍尔斯和$δ$ neighter-nighborhoods of the curves的发病率和$δ$ neightoborhoods之间的发病率进行了改进。这些证明取决于将措施的新多尺度分解为“霜冻作品”，这可能引起了独立的关注。
We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to $1$ in $\mathbb{R}^2$ or close to $3/2$ in $\mathbb{R}^3$, then for $y\in A$ outside of a very sparse set, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff dimension at least $1/2+c$, where $c$ is universal. Furthermore, the same holds if the distances are taken with respect to a $C^2$ norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between $δ$-balls and $δ$-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into ``Frostman pieces'' that may be of independent interest.
