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Self-similar measures and the Rajchman property
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For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erd{\"o}s [2] and Salem [12]. We prove weak forms of their results for general self-similar measures associated to affine contractions of the real line.
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Purity results for some arithmetically defined measures
For Pisot bases and sofic digit shifts, generalized Erdős measures are always pure, and absolute continuity is equivalent to vanishing of the Fourier limits along beta powers.
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