Harmonic analysis, additive combinatorics and geometric measure theory

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Fractals are objects that exhibit a multiscale structure and can be assigned a "number of dimensions" that is not necessarily integer. They are ubiquitous in nature (plant growth, landscape formation), and can also be produced by human activity (communication systems, infrastructure networks). In mathematics, they arise in many problems in fields such as mathematical physics or partial differential equations, and are studied extensively in geometric measure theory and dynamical systems. Fractals often display chaotic behaviour, in the sense that a fractal object constructed by iterating a relatively simple replication rule can exhibit very high levels of complexity and behave in ways that are very difficult to predict in advance. The first goal of the proposed research is to develop new methods of studying such structures, based on harmonic analysis and additive combinatorics. Traditionally, the use of harmonic analysis in the study of fractals has been relatively limited. I plan to develop a theory of singular and oscillatory integrals for fractal sets, modelled after the analogous theory for smooth manifolds in classical harmonic analysis (including restriction estimates, maximal and averaging operators, and differentiation theorems), but also incorporating new features that only occur in the fractal setting. This work will draw on the insights and methods from both harmonic analysis and the more recently developed field of additive combinatorics. While some foundational results have already been obtained, proving that this is a viable and interesting theory, many further questions remain open. In particular, fractal sets can exhibit harmonic-analytic behaviour that does not have exact analogues either for manifolds or for the discrete objects studied in additive combinatorics, and I would like to investigate these new phenomena.The second goal is to apply these developments to questions in dimension theory, geometric measure theory and dynamical systems. A central family of questions in geometric measure theory concerns projections, slices, intersections, and arithmetic sums and differences of sets of specified dimensionality. Of particular significance are fractal sets that arise in dynamical systems and mathematical physics, such as attractors or invariant sets. For example, it can be very easy to say what a "typical" projection or slice of a fractal should look like, but very difficult to prove a similar statement about a specific projection or slice. Similarly, there are many situations where qualitative results are easy to prove, but it is much harder to obtain a quantitative estimate. Harmonic analysis has been useful in such contexts in the past, and I expect that the new methods I plan to develop will lead to further advances.