Fourier decay and absolute continuity for typical homogeneous self-similar measures in ${\mathbb{R}}^d$ for $d\ge 3$

Abstract

We consider iterated function systems (IFS) in ${\mathbb{R}}^d$ for $d\ge 3$ of the form $\{f_j(x)=\lambda \mathcal{O}x+a_j{\}}_{j=0}^m$, with $a_0=0$ and $m\ge 1$. Here $\lambda \in (0,1)$ is the contraction ratio and $\mathcal{O}$ is an orthogonal matrix. Given a positive probability vector $p$, there is a unique invariant (stationary) measure for the IFS, called (in this case) a homogeneous self-similar measure, which we denote $\mu (\lambda \mathcal{O},\mathcal{D},p)$, where $\mathcal{D}=\{a_0,\dots ,a_m\}$ is the set of ``vector digits''. We obtain two results on Fourier decay for such measures. First we show that if $\mathcal{D}$ spans ${\mathbb{R}}^d$, then for every fixed $\mathcal{O}$ and $p$ the measure $\mu (\lambda \mathcal{O},\mathcal{D},p)$ has power Fourier decay (equivalently, positive Fourier dimension) for all but a zero-Hausdorff dimension set of $\lambda$. In our second result we do not impose any restrictions on $\mathcal{D}$, other than the necessary one of affine irreducibility, and obtain power Fourier decay for almost all homogeneous self-similar measures; however, only for even $d\ge 4$. Combined with recent work of Corso and Shmerkin [arXiv:2409.04608] , these results imply absolute continuity for almost all self-similar measures under the same assumptions, in the super-critical parameter region.

25 pages