Abstract
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$ dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg’s long-standing conjecture on the dimension of the intersections of $\times p$- and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions.
The proof of the main result is inspired by M. Hochman’s approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman’s theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
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author = {Katznelson, Yitzhak and Weiss, Benjamin},
title = {A simple proof of some ergodic theorems},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {42},
year = {1982},
number = {4},
pages = {291--296},
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doi = {10.1007/BF02761409},
url = {https://doi.org/10.1007/BF02761409},
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author = {Peres, Yuval and Schlag, Wilhelm and Solomyak, Boris},
title = {Sixty years of {B}ernoulli convolutions},
booktitle = {Fractal Geometry and Stochastics, {II}},
venue = {{G}reifswald/{K}oserow, 1998},
series = {Progr. Probab.},
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pages = {39--65},
publisher = {Birkhäuser, Basel},
year = {2000},
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mrnumber = {1785620},
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JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {32},
YEAR = {2019},
NUMBER = {2},
PAGES = {351--397},
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MRCLASS = {28A80 (42A85)},
MRNUMBER = {3904156},
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FJOURNAL = {Annals of Mathematics. Second Series},
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note = {to appear},
arxiv = {1609.08053},
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url = {https://doi.org/10.4007/annals.2019.189.3.2},
}