Abstract
We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p \mod 1$ and $\times q \mod 1$, respectively, and if $\log p/\log q\notin \Bbb{Q}$, then for all real numbers $u$ and $v$,
\[
\dim_{\rm H}(uA+v)\cap B\le \max\{0,\dim_{\rm H}A+\dim_{\rm H}B-1\}.
\]
We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on $\Bbb{R}$. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.
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author = {Shmerkin, Pablo and Suomala, Ville},
title = {Spatially independent martingales, intersections, and applications},
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doi = {10.1090/memo/1195},
url = {https://doi.org/10.1090/memo/1195},
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title = {Conditional measures and conditional expectation; {R}ohlin's disintegration theorem},
journal = {Discrete Contin. Dyn. Syst.},
fjournal = {Discrete and Continuous Dynamical Systems. Series A},
volume = {32},
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issn = {1078-0947},
mrclass = {28A50 (28C05 28C15)},
mrnumber = {2900561},
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author = {Sinai, {\relax Ya} G.},
title = {A weak isomorphism of transformations with invariant measure},
journal = {Dokl. Akad. Nauk SSSR},
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volume = {147},
year = {1962},
pages = {797--800},
issn = {0002-3264},
mrclass = {28.70},
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number = {4},
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author = {Sinai, {\relax Ya} G.},
title = {On a weak isomorphism of transformations with invariant measure},
journal = {Mat. Sb. (N.S.)},
volume = {63 (105)},
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pages = {23--42},
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}
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