"Harmonic analysis, additive combinatorics and geometric measure theory"

欧几里德调和分析的一个基本特征是分析与几何之间的相互作用。例如，n维空间中超曲面的几何特征，特别是曲率和光滑性，反映在与这些曲面相关的分析估计（限制，最大估计）的范围内。这一问题已得到广泛和深入的研究，但许多基本问题仍然悬而未决。 最近，类似的现象也被观察到分形集。这甚至可以发生在维度1中，其中没有低维表面，但有许多分形集。事实证明，“随机”分形通常表现得像弯曲的超曲面，如球体，而分形表现出算术结构（例如，几何结构）。中间三分之一康托集）表现得像平面。这反过来又涉及到经典问题的几何测度论有关的预测和交叉家庭的分形集。 我建议使用调和分析和加法组合学的组合方法来研究这些现象。对整数集合中的“随机性”和“算术结构”的研究是最近重大进展的关键部分，例如关于素数中长算术级数的格林-陶定理。将这些强大的方法和见解转移到测度论的环境中，可能会对超曲面和分形集的分析性质产生新的影响。

A fundamental feature of Euclidean harmonic analysis is the interplay between analysis and geometry. For instance, the geometric features of hypersurfaces in an n-dimensional space, especially curvature and smoothness, are reflected in the ranges of analytic estimates (restriction, maximal estimates) associated with such surfaces. The subject has been studied extensively and in depth, but many fundamental questions still remain open. More recently, similar phenomena have also been observed for fractal sets. This can occur even in dimension 1, where there are no lower-dimensional surfaces but many fractal sets. It turns out that "random" fractals often behave like curved hypersurfaces such as spheres, whereas fractals exhibiting arithmetic structure (eg. the middle-third Cantor set) behave like flat surfaces. This is in turn related to classical questions of geometric measure theory concerning projections and intersections of families of fractal sets. I propose to study these phenomena using combined methods from harmonic analysis and additive combinatorics. The study of "randomness" and "arithmetic structure" in sets of integers was a key part of recent major advances such as the Green-Tao theorem on long arithmetic progressions in the primes. Transferring these powerful methods and insights to a measure-theoretic setting is likely to shed a new light on the analytic properties of both hypersurfaces and fractal sets.