Abstract
We present a self-contained proof of a formula for the \(L^q\) dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the \(L^q\) norm of convolutions). This is a special case of a more general result of the author from Shmerkin (Ann Math, 2019), and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.
Partially supported by Projects CONICET-PIP 11220150100355 and PICT 2015-3675 (ANPCyT).
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Shmerkin, P. (2019). \(L^q\) Dimensions of Self-similar Measures and Applications: A Survey. In: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U. (eds) New Trends in Applied Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-32353-0_9
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DOI: https://doi.org/10.1007/978-3-030-32353-0_9
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