Fourier decay for homogeneous self-affine measures

TL;DR

This paper proves that for almost all parameter choices, homogeneous self-affine measures in Euclidean space exhibit power decay in their Fourier transforms, revealing a generic regularity property of such measures.

Contribution

It establishes the Fourier decay property for a broad class of self-affine measures with homogeneous, diagonal linear parts, extending understanding of their harmonic analysis behavior.

Findings

Almost all parameter tuples lead to Fourier decay

Self-affine measures have power Fourier decay at infinity

Results hold for non-degenerate, homogeneous systems

Abstract

We show that for Lebesgue almost all $d$-tuples $(\theta_1,\ldots,\theta_d)$, with $|\theta_j|>1$, any self-affine measure for a homogeneous non-degenerate iterated function system $\{Ax+a_j\}_{j=1}^m$ in ${\mathbb R}^d$, where $A^{-1}$ is a diagonal matrix with the entries $(\theta_1,\ldots,\theta_d)$, has power Fourier decay at infinity.