Abstract: Determining when the Fourier transform of a measure decays to zero as a function of the frequency, and estimating the speed of decay if so, is an important problem. We will discuss this problem in relation to fractal measures arising from iterated function systems, explaining that systems with non-linearity often result in good decay. In particular, in joint work with Simon Baker, we have used a disintegration technique to prove that the Fourier transforms of non-linear pushforwards of a general class of fractal measures decay at a polynomial rate. Combining this with a result of Algom – Rodriguez Hertz – Wang and Baker – Sahlsten, we prove that for any IFS on the line consisting of analytic contractions, at least one of which is not affine, every non-atomic self-conformal measure exhibits polynomial Fourier decay. This has several interesting applications; one application relates the presence of normal numbers in fractal sets.
