Title: Quantitative Fourier decay of nonlinear images of self-similar measures
Abstract: In 1984, Kaufman observed that although the Fourier coefficients of the natural measure on the middle-third Cantor set do not tend to zero, nonlinear perturbations of this measure do exhibit Fourier decay. Kaufman’s result has in recent years been generalised significantly: under mild assumptions, nonlinear images of self-similar measures on R^n have polynomial Fourier decay. In this talk, we will describe some of these generalisations, with particular emphasis on quantifying the rate of decay, where tools such as fractal uncertainty principles can be used. As one of several applications, we make progress towards the following challenging problem: what Hausdorff dimension thresholds guarantee that arithmetic products of self-similar sets on the line have positive measure (or non-empty interior)? This talk is mostly based on recent joint work with Han Yu.
