Assouad维数的非线性投影定理及其应用
A nonlinear projection theorem for Assouad dimension and applications
AI总结：
本文证明了Assouad维数的一般非线性投影定理，应用于距离集、径向投影和和积现象，完全解决了平面距离集问题，并给出了尖锐估计。
摘要：
我们证明了关于Assouad维数的一般非线性投影定理。该定理在距离集、径向投影和和积现象等多个领域具有应用。在距离集的设定下，我们能够完全解决Assouad维数的平面距离集问题，既处理了棘手的“临界情形”，又为Assouad维数小于1的集合提供了尖锐的估计。在高维设定下，我们将该问题与相关（正交）投影定理中例外集的维数联系起来。我们还获得了关于固定距离集的结果，并且当距离相对于充分弯曲的范数取定时，我们的结果仍然成立。作为另一个应用，我们证明了关于Assouad维数的径向投影定理，并对例外集的Hausdorff维数给出了尖锐的估计。
英文摘要：
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.
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