Fourier transform of nonlinear images of self-similar measures: quantitative aspects

Abstract

This paper relates to the Fourier decay properties of images of self-similar measures $\mu$ 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 $\mu$ commute and the graph of $f$ has nonvanishing Gaussian curvature, then the Fourier dimension of the image measure is at least $\max\left\{ \frac{2(2\kappa_2 - k)}{4 + 2\kappa_* - k} , 0 \right\}$, where $\kappa_2$ is the lower correlation dimension of $\mu$ and $\kappa_*$ is the Assouad dimension of the support of $\mu$. Under some additional assumptions on $\mu$, 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.