## Abstract

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
_{j}f
_{j}μ, ∑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}.

Original language | English |
---|---|

Article number | 107349 |

Journal | Advances in Mathematics |

Volume | 374 |

Early online date | 10 Aug 2020 |

DOIs | |

Publication status | Published - 18 Nov 2020 |