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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P. Shmerkin and V. Suomala, "Spatially independent martingales, intersections, and applications," Mem. Amer. Math. Soc., vol. 251, iss. 1195, p. v, 2018.
@ARTICLE{SS2015,
author = {Shmerkin, Pablo and Suomala, Ville},
title = {Spatially independent martingales, intersections, and applications},
journal = {Mem. Amer. Math. Soc.},
fjournal = {Memoirs of the American Mathematical Society},
volume = {251},
year = {2018},
number = {1195},
pages = {v+102},
issn = {0065-9266},
isbn = {978-1-4704-2688-0; 978-1-4704-4264-4},
mrclass = {60D05 (28A75 28A78 28A80 42A38 60G46 60G57)},
mrnumber = {3756896},
mrreviewer = {Elena Villa},
doi = {10.1090/memo/1195},
url = {https://doi.org/10.1090/memo/1195},
zblnumber = {07000048},
} -
[Simmons2012]
D. Simmons, "Conditional measures and conditional expectation; Rohlin’s disintegration theorem," Discrete Contin. Dyn. Syst., vol. 32, iss. 7, pp. 2565-2582, 2012.
@ARTICLE{Simmons2012,
author = {Simmons, David},
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},
year = {2012},
number = {7},
pages = {2565--2582},
issn = {1078-0947},
mrclass = {28A50 (28C05 28C15)},
mrnumber = {2900561},
doi = {10.3934/dcds.2012.32.2565},
url = {https://doi.org/10.3934/dcds.2012.32.2565},
zblnumber = {1244.28002},
} -
[Sinai62]
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@ARTICLE{Sinai62,
author = {Sinai, {\relax Ya} G.},
title = {A weak isomorphism of transformations with invariant measure},
journal = {Dokl. Akad. Nauk SSSR},
fjournal = {Doklady Akademii Nauk SSSR},
volume = {147},
year = {1962},
pages = {797--800},
issn = {0002-3264},
mrclass = {28.70},
mrnumber = {0161960},
number = {4},
url = {http://mi.mathnet.ru/eng/dan/v147/i4/p797},
} -
[Sinai64]
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@ARTICLE{Sinai64,
author = {Sinai, {\relax Ya} G.},
title = {On a weak isomorphism of transformations with invariant measure},
journal = {Mat. Sb. (N.S.)},
volume = {63 (105)},
year = {1964},
pages = {23--42},
mrclass = {28.70},
mrnumber = {0161961},
mrreviewer = {J. Auslander},
number = {1},
note={[English translation: \emph{Amer. Math. Soc. Transl.} {\bf 57} (1966), 123--143]},
url = {http://mi.mathnet.ru/eng/msb/v105/i1/p23},
} -
[Waltersbook] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982, vol. 79.
@BOOK{Waltersbook,
author = {Walters, Peter},
title = {An Introduction to Ergodic Theory},
series = {Grad. Texts in Math.},
volume = {79},
publisher = {Springer-Verlag, New York},
year = {1982},
pages = {ix+250},
isbn = {0-387-90599-5},
mrclass = {28Dxx (54H20 58F11)},
mrnumber = {0648108},
mrreviewer = {M. A. Akcoglu},
zblnumber = {0958.28011},
}