Fourier decay for homogeneous self-affine measures

Abstract

lim p μpξq “ 0, as |ξ| Ñ 8, where |ξ| is a norm (say, the Euclidean norm) of ξ P Rd. Whereas absolutely continuous measures are Rajchman by the Riemann-Lebesgue Lemma, it is a subtle question to decide which singular measures are such, see, e.g., the survey of Lyons [14]. A much stronger property, useful for many applications is the following. Definition 1.1. For α ą 0 let Ddpαq “ ν finite positive measure on Rd : |p νptq| “ Oνp|t|q, |t| Ñ 8 (