Polynomial Fourier decay for fractal measures and their pushforwards

TL;DR

The paper develops a general framework to obtain polynomial Fourier decay for pushforwards $F\mu$ of fractal measures under nonlinear maps, covering broad CIFS/fibre-product settings and allowing analytic or $C^2$ nonlinearities with nonvanishing second derivatives in appropriate directions. The authors introduce a disintegration-based strategy combined with large deviations and Erdős–Kahane-type arguments to prove decay outside sparse frequencies, and then extend these results to nonlinear images via Taylor expansion and Frostman-type estimates. Key results include a general non-linear pushforward theorem (Theorem $t$:main) and an analytic pushforward theorem, together with corollaries for self-similar/self-conformal measures; the work yields applications to Fourier uniqueness, Fractal Uncertainty Principles, Fourier restriction, normal numbers, equidistribution, and conditional mixing. The approach unifies and extends previous results without requiring homogeneity or strong separation, and offers a versatile toolkit for harmonic analysis on fractal measures with wide implications in dynamics and number theory.

Abstract

We prove that the pushforwards of a very general class of fractal measures $μ$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,η>0$ such that $|\widehat{Fμ}(ξ)|\leq C|ξ|^{-η}$ for all $ξ\neq 0$. Using this, we prove that if $Φ= \{ \varphi_a \colon [0,1] \to [0,1] \}_{a \in \mathcal{A}}$ is an iterated function system consisting of analytic contractions, and there exists $a \in \mathcal{A}$ such that $\varphi_a$ is not an affine map, then every non-atomic self-conformal measure for $Φ$ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets.

Figures (1)

• Figure 1: Let \Phi_{1}=\{\varphi_{1}(x,y)=(\frac{x}{2},\frac{y}{3}),\, \varphi_{2}(x,y)=(\frac{x}{2},\frac{y+2}{3}),\varphi_{3}(x,y)=(\frac{x+1}{2},\frac{y+1}{3})\},\Phi_{2}=\{\varphi_{1}(x,y)=(\frac{x}{3},\frac{y}{5}),\, \varphi_{2}(x,y)=(\frac{x}{3},\frac{4y+5}{10}),\varphi_{3}(x,y)=(\frac{x+1}{2},\frac{y+1}{2}),\, \varphi_{4}(x,y)=(\frac{x+2}{5},\frac{9y}{10}),\varphi_{5}(x,y)=(\frac{7x+1}{10},\frac{3y+6}{10}),\, \varphi_{6}(x,y)=(\frac{3x+6}{10},\frac{2x+2}{5})\},\Phi_{3}=\{\varphi_{1}(x,y)=(\frac{x}{3},\frac{y}{5}),\, \varphi_{2}(x,y)=(\frac{x+1}{3},\frac{y+2}{5}),\varphi_{3}(x,y)=(\frac{x+2}{3},\frac{y+4}{5})\}.The attractor for $\Phi_{1}$ is a Bedford--McMullen carpet to which Theorem \ref{['t:main']} can be applied, and $\Phi_{2}$ is an overlapping IFS to which Theorem \ref{['t:main']} applies. Notice that for both $\Phi_{1}$ and $\Phi_{2}$ the maps $\varphi_{1}$ and $\varphi_{2}$ have the same horizontal component but have vertical components with distinct fixed points. Therefore $\Phi_{1}$ and $\Phi_{2}$ can both be realised as fibre product CIFSs. Notice however that $\Phi_{3}$ cannot be realised as a fibre product CIFS because the vertical slices through this set always consist of singletons.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Corollary 1.5
• proof
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof