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 Erdos and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.
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Received by the editors July 30, 2019, and, in revised form, September 22, 2020. 2020 Mathematics Subject Classification. Primary 28A80, 42A16, 60G18. This research was supported by the Israel Science Foundation Grant 396/15.
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