Fourier transform of self-affine measures

Abstract

Suppose F is a self-affine set on Rd, d≥2, which is not a singleton, associated to affine contractions fj=Aj+bj, Aj∈GL(d,R), bj∈Rd, j∈A, for some finite A. We prove that if the group Γ generated by the matrices Aj, j∈A, forms a proximal and totally irreducible subgroup of GL(d,R), then any self-affine measure μ=∑pjfjμ, ∑pj=1, 0<pj<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 Lp 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 Sd−1.