Fourier decay of self-similar measures and self-similar sets of uniqueness

Abstract

In this paper, we investigate the Fourier transform of self-similar measures on R. We provide quantitative decay rates of Fourier transform of some self-similar measures. Our method is based on random walks on lattices and Diophantine approximation in number fields. We also completely identify all self-similar sets which are sets of uniqueness. This generalizes a classical result of Salem and Zygmund.

16 pages, this is the final accepted version to appear in Ann. PDE, revision based on referee's comments, added assumption in Corollary 1.6 that r^-1 is an algebraic integer

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