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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.
@article{10_4007_annals_2019_189_3_2,
author = {Meng Wu},
title = {A proof of {Furstenberg{\textquoteright}s} conjecture on the intersections of $\times p$- and $\times q$-invariant sets},
journal = {Annals of mathematics},
pages = {707--751},
publisher = {mathdoc},
volume = {189},
number = {3},
year = {2019},
doi = {10.4007/annals.2019.189.3.2},
mrnumber = {3961082},
zbl = {07097489},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.2/}
}
TY - JOUR AU - Meng Wu TI - A proof of Furstenberg’s conjecture on the intersections of $\times p$- and $\times q$-invariant sets JO - Annals of mathematics PY - 2019 SP - 707 EP - 751 VL - 189 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.2/ DO - 10.4007/annals.2019.189.3.2 LA - en ID - 10_4007_annals_2019_189_3_2 ER -
%0 Journal Article %A Meng Wu %T A proof of Furstenberg’s conjecture on the intersections of $\times p$- and $\times q$-invariant sets %J Annals of mathematics %D 2019 %P 707-751 %V 189 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.2/ %R 10.4007/annals.2019.189.3.2 %G en %F 10_4007_annals_2019_189_3_2
Meng Wu. A proof of Furstenberg’s conjecture on the intersections of $\times p$- and $\times q$-invariant sets. Annals of mathematics, Tome 189 (2019) no. 3, pp. 707-751. doi: 10.4007/annals.2019.189.3.2
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