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Fourier decay for self-similar measures

We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erdős and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.

preprint2020arXivOpen access
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