Fourier asymptotics of statistically self-similar measures

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Abstract

In this paper we investigate the pointwise Fourier decay of some selfsimilar random measures. As an application we construct statistically selfsimilar Salem sets. For example, our result shows that a “slight” random perturbation of the classical Cantor set becomes a “nice” set in the sense that its Fourier dimension equals its Hausdorff dimension.

Communicated by Robert S. Strichartz