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

This paper relates to the Fourier decay properties of images of self-similar measures $\mu$ on ${\mathbb{R}}^k$ under nonlinear smooth maps $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^k\to \mathbb{R}$. 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 $\max \{\frac{2(2{\kappa }_2-k)}{4+2{\kappa }_{\ast }-k},0\}$, where ${\kappa }_2$ is the lower correlation dimension of $\mu$ and ${\kappa }_{\ast }$ 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 ${\dim }_{\mathrm{H}}F>(\sqrt{65}-5)\mathrm{/}4=0.765\dots$ then the arithmetic product set $F\cdot F=\{xy:x,y\in F\}$ has positive Lebesgue measure, while if ${\dim }_{\mathrm{H}}F>(-3+\sqrt{41})\mathrm{/}4=0.850\dots$ 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.