Fourier transform of self-affine measures

Jialun Li, Tuomas Sahlsten

Research output: Contribution to journalArticleScientificpeer-review

18 Citations (Scopus)

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 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.

Original languageEnglish
JournalAdvances in Mathematics
DOIs
Publication statusPublished - 18 Nov 2020
MoE publication typeA1 Journal article-refereed

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