Fourier transform of nonlinear images of self-similar measures: quantitative aspects
Amlan Banaji, Han Yu
TL;DR
This work establishes polynomial Fourier decay for nonlinear images of self-similar measures under $C^2$ maps with graphs of nonzero Gaussian curvature, deriving explicit decay exponents in terms of fractal dimensions such as $κ_2$, $κ_*$, $d_{∞}$, and $κ_1$. The authors develop a framework that lifts measures to graph space, decomposes and linearizes the Fourier transform, and leverages Hölder estimates and $L^p$-dimension bounds; the homogeneous/non-expanding cases are treated in detail. They further apply these decay results to nonlinear arithmetic of self-similar sets on the line, proving dimension-based thresholds ensuring positive Lebesgue measure for $E\cdot F$ and nonempty interiors for $E\cdot F\cdot G$. The fractal uncertainty principle is employed to sharpen exponents in favorable settings, and the paper discusses extensions to quadratic and holomorphic maps, as well as open questions about optimal bounds and broader implications. Overall, the work connects fractal geometry, harmonic analysis, and additive/m multiplicative structure to yield quantitative Fourier-decay results with applications to nonlinear arithmetic of fractal sets.
Abstract
This paper relates to the Fourier decay properties of images of self-similar measures $μ$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the similarities defining $μ$ commute and the graph of $f$ has nonvanishing Gaussian curvature, then the Fourier dimension of the image measure is at least $\max\left\{ \frac{2(2κ_2 - k)}{4 + 2κ_* - k} , 0 \right\}$, where $κ_2$ is the lower correlation dimension of $μ$ and $κ_*$ is the Assouad dimension of the support of $μ$. Under some additional assumptions on $μ$, we use recent breakthroughs in the fractal uncertainty principle to obtain further improvements for the decay exponents. We give several applications to nonlinear arithmetic of self-similar sets $F$ in the line. For example, we prove that if $\dim_{\mathrm H} F > (\sqrt{65} - 5)/4 = 0.765\dots$ then the arithmetic product set $F \cdot F = \{ xy : x,y \in F \}$ has positive Lebesgue measure, while if $\dim_{\mathrm H} F > (-3 + \sqrt{41})/4 = 0.850\dots$ then $F \cdot F \cdot F$ has non-empty interior. One feature of the above results is that they do not require any separation conditions on the self-similar sets.