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 ${\mathcal D} = \{a_0,\ldots,a_m\}$ 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.