Fourier transform of self-affine measures

Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, 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 $Γ$ 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 $μ= \sum p_j f_j μ$, $\sum p_j = 1$, $0 < p_j < 1$, $j \in \mathcal{A}$, on $F$ is a Rajchman measure: the Fourier transform $\widehatμ(ξ) \to 0$ as $|ξ| \to \infty$. 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 $Γ$ is connected real split Lie group in the Zariski topology, then $\widehatμ(ξ)$ has a power decay at infinity. Hence $μ$ is $L^p$ improving for all $1 < p < \infty$ and $F$ has positive Fourier dimension. In dimension $d = 2,3$ the irreducibility of $Γ$ and non-compactness of the image of $Γ$ in $\mathrm{PGL}(d,\mathbb{R})$ is enough for power decay of $\widehatμ$. The proof is based on quantitative renewal theorems for random walks on the sphere $\mathbb{S}^{d-1}$.

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