Fourier decay for nonlinear fractal measures

ongoing

This MSCA project studies the Fourier transform of dynamically defined fractal measures. Determining whether the Fourier transform decays to zero, and if so then at what rate, provides important information about the measure. There has been an explosion of interest in this topic in the past few years, spurred by a wealth of applications: Fourier decay is connected to a wide range of problems from harmonic analysis, Diophantine approximation, additive combinatorics, and quantum chaos. Up to now, most of the literature has focused on the Fourier decay of self-similar measures in dimensions one and two. The aim of this project is to build a theory covering measures generated by non-linear dynamics, and in arbitrary dimensions. One of the main objectives is to use novel transfer operator methods to characterise when measures generated by non-linear analytic conformal dynamics have power Fourier decay. Another goal is to make substantial improvements to the bounds for the decay exponent in many cases. The new Fourier decay estimates will be applied to establish fractal analogues of fundamental theorems of Khintchine and Gallagher in Diophantine approximation, and obtain new fractal uncertainty principles and Fourier restriction estimates in harmonic analysis. The researcher (PhD 2023) already has a track record of multiple cutting-edge works on the theory of Fourier decay of fractal measures, and on the dimension theory of iterated function systems. He therefore has the ideal blend of expertise in both fractal geometry and Fourier transforms to complete the ambitious objectives. The project will take place at the University of Jyväskylä in the group of T. Orponen, an ERC grantee working at the interface of fractal geometry and harmonic analysis.