Fourier transform of self-affine measures

Suppose F is a self-affine set on R ^d, d≥2, which is not a singleton, associated to affine contractions f _j=A _j+b _j, A _j∈GL(d,R), b _j∈R ^d, j∈A, for some finite A. We prove that if the group Γ generated by the matrices A _j, j∈A, forms a proximal and totally irreducible subgroup of GL(d,R), then any self-affine measure μ=∑p _jf _jμ, ∑p _j=1, 0<p _j<1, j∈A, on F is a Rajchman measure: the Fourier transform μˆ(ξ)→0 as |ξ|→∞. 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 μˆ(ξ) has a power decay at infinity. Hence μ is L ^p improving for all 1<p<∞ and F has positive Fourier dimension. In dimension d=2,3 the irreducibility of Γ and non-compactness of the image of Γ in PGL(d,R) is enough for power decay of μˆ. The proof is based on quantitative renewal theorems for random walks on the sphere S ^d−1.