RUI: Fourier Restriction and Fourier Dimension for Fractals

Many objects in nature are self-similar, that is, they display similar features when examined at different scales. Examples include the shapes of river channels, snowflakes, ferns, and lightning, just to name a few. In mathematics, these self-similar objects are characterized as “fractals.” Fractals appear in the study of paths of particle motion, branching biological structures, crystals, turbulence, cloud formation, and other areas. The Fourier transform is a mathematical operation ubiquitous in signal and image processing, spectroscopy, quantum mechanics, and many other fields of science and engineering. The overarching theme of this research project is the study of the connections between the (geometric and arithmetic) structure of fractal sets and the Fourier transforms of functions and measures defined on such sets. The work will be done in collaboration with undergraduate and graduate students, in keeping with the goal of the PI to help recruit and train future mathematicians and other STEM professionals. The PI will solve problems in the construction of fractal Salem sets (i.e., sets whose Hausdorff and Fourier dimensions are equal), calculation of the exact Fourier dimension of fractal sets, optimality of Fourier restriction on fractals, and construction of measures with rapid Fourier decay on various fractal sets (including fractal subsets of curves and surfaces, sets of badly approximable vectors, and sums and products of fractal sets). To solve these problems, the PI will extend techniques he has developed over the decade in solving related problems. Progress will feedback into other areas of analysis by providing insight to researchers working on problems in geometric measure theory, additive combinatorics, and partial differential equations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.