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

TL;DR

This paper studies the Fourier decay properties of homogeneous self-similar measures in higher dimensions, showing that under certain conditions, these measures exhibit power decay and are absolutely continuous for most parameters.

Contribution

It establishes new results on Fourier decay and absolute continuity of self-similar measures in ${f R}^d$ for $d \\ge 3$, extending previous work to higher dimensions and broader conditions.

Findings

Power Fourier decay holds for measures with spanning digit sets outside a zero-Hausdorff dimension set of contraction ratios.

Almost all homogeneous self-similar measures exhibit power Fourier decay under affine irreducibility, for even dimensions $d \\ge 4$.

Results imply absolute continuity of these measures in the super-critical parameter region, combining with recent work.

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.