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) is an element of GL(d, R), b(j) is an element of R-d, j is an element of A, for some finite A. We prove that if the group Gamma generated by the matrices A(j), j is an element of A, forms a proximal and totally irreducible subgroup of GL(d, R), then any self-affine measure mu = Sigma p(j)f(j)mu, Sigma p(j) = 1, 0 < p(j) < 1, j is an element of A, on F is a Rajchman measure: the Fourier transform (mu) over cap(xi) -> 0 as vertical bar xi vertical bar -> infinity. 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 (mu) over cap(xi) has a power decay at infinity. Hence mu is L-p improving for all 1 < p < infinity and F has positive Fourier dimension. In dimension d = 2, 3 the irreducibility of Gamma and non-compactness of the image of Gamma in PGL(d, R) is enough for power decay of (mu) over cap. The proof is based on quantitative renewal theorems for random walks on the sphere Sd-1. (C) 2020 Elsevier Inc. All rights reserved.