Fourier transform of self-affine measures

Abstract

Suppose $F$ is a self-affine set on ${\mathbb{R}}^d$, $d\ge 2$, which is not a singleton, associated to affine contractions $f_j=A_j+b_j$, $A_j\in \mathrm{GL}(d,\mathbb{R})$, $b_j\in {\mathbb{R}}^d$, $j\in \mathcal{A}$, for some finite $\mathcal{A}$. We prove that if the group $\Gamma$ generated by the matrices $A_j$, $j\in \mathcal{A}$, forms a proximal and totally irreducible subgroup of $\mathrm{GL}(d,\mathbb{R})$, then any self-affine measure $\mu =\sum p_j{f}_j\mu$, $\sum p_j=1$, $0<p_j<1$, $j\in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\hat{\mu }(\xi )\to 0$ as $|\xi |\to \infty$. As an application this shows that self-affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series. Moreover, if the Zariski closure of $\Gamma$ is connected real split Lie group in the Zariski topology, then $\hat{\mu }(\xi )$ has a power decay at infinity. Hence $\mu$ is $L^p$ improving for all $1<p<\infty$ and $F$ has positive Fourier dimension. In dimension $d=2,3$ the irreducibility of $\Gamma$ and non-compactness of the image of $\Gamma$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\hat{\mu }$. The proof is based on quantitative renewal theorems for random walks on the sphere ${\mathbb{S}}^{d-1}$.

v2: 27 pages, updated references. Accepted to Advances in Math