论文标题
Assouad维度和应用的非线性投影定理
A nonlinear projection theorem for Assouad dimension and applications
论文作者
论文摘要
我们证明了Assouad维度的一般非线性投影定理。该定理具有多种应用，包括距离集，径向投影和总产品现象。在距离集的设置中，我们能够完全解决Assouad维度的平面距离集问题，既处理了尴尬的“关键情况”，又为Assouad维度小于1的设置提供了尖锐的估计。在较高的维度设置中，我们将问题连接到相关（Orthogogonal）投影neorem中的除外的问题。我们还获得了固定距离集的结果，当相对于足够弯曲的规范采取距离时，我们的结果仍然存在。作为另一个应用，我们证明了对Assouad维度的径向投影定理，对特殊集合的Hausdorff维度进行了锐利的估计。
We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.
