Self-similar and self-conformal measures with slow Fourier decay
Simon Baker, Amlan Banaji
TL;DR
The paper shows that Rajchman self-similar and nonlinear self-conformal measures can exhibit arbitrarily slow Fourier decay, answering negative questions about universal polynomial or logarithmic decay. It develops a framework for both constructing generic slow-decay self-similar measures via a parameter family and producing $C^{ abla}$ pushforwards with slow decay, including explicit Liouville-type examples and $C^{ abla}$-smooth IFSs whose second-derivative structure yields slow decay along subsequences. A key contribution is the demonstration that slow Fourier decay persists even under nonlinear, smooth transformations, and that such measures can be Rajchman while still supporting very slow equidistribution properties in dynamical systems such as base-10 expansions. The work provides explicit constructions, higher-dimensional generalizations, and a blend of harmonic analysis, fractal geometry, and Diophantine techniques (e.g., Brémont-type criteria) to realize these counterintuitive phenomena. Its results have implications for fractal uncertainty principles, normality in expansions, and the broader understanding of how smooth perturbations interact with fractal measures in Fourier space.
Abstract
Given any function $φ\colon [0,\infty)\to (0,1]$ satisfying $\lim_{ξ\to\infty}φ(ξ) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehatμ$ satisfies \[ \limsup_{ξ\to\infty}\frac{|\widehatμ(ξ)|}{φ(ξ)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $μ$ such that $μ$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x \mod 1)_{n=1}^{\infty}$ equidistributes extremely slowly.