"Harmonic analysis, additive combinatorics and geometric measure theory"

欧几里得调和分析的一个基本特征是分析和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，反映在与此类曲面相关的解析估计(约束、最大估计)的范围内。这个主题已经得到了广泛和深入的研究，但许多基本问题仍然悬而未决。 最近，在分形集上也观察到了类似的现象。即使在维度1中也可能发生这种情况，在维度1中没有较低维度的曲面，但有许多分形集。事实证明，“随机”的分形图通常表现得像球体等弯曲的超曲面，而分形图则表现出算术结构(例如。中间三分之一的康托集)的行为类似于平面。这反过来又涉及到几何测度论中关于分形族的投影和交集的经典问题。 我建议用调和分析和加性组合数学相结合的方法来研究这些现象。对整数集的“随机性”和“算术结构”的研究是最近重大进展的关键部分，例如关于素数中的长算术级数的Green-Tao定理。将这些强大的方法和见解转移到测度论的背景下，可能会为超曲面和分形集的分析性质带来新的曙光。

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.