2022, 18: 1-11. doi: 10.3934/jmd.2022001

A new dynamical proof of the Shmerkin–Wu theorem

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555, USA

Received  October 09, 2020 Revised  September 29, 2021 Published  December 2021

Let $ a < b $ be multiplicatively independent integers, both at least $ 2 $. Let $ A,B $ be closed subsets of $ [0,1] $ that are forward invariant under multiplication by $ a $, $ b $ respectively, and let $ C : = A\times B $. An old conjecture of Furstenberg asserted that any planar line $ L $ not parallel to either axis must intersect $ C $ in Hausdorff dimension at most $ \max\{\dim C,1\} - 1 $. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.

Citation: Tim Austin. A new dynamical proof of the Shmerkin–Wu theorem. Journal of Modern Dynamics, 2022, 18: 1-11. doi: 10.3934/jmd.2022001
References:
[1]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in Ergodic Theory and Topological Dynamics., Book draft, available online at https://tbward0.wixsite.com/books/entropy. Google Scholar

[2]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014.  Google Scholar

[3]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[4]

H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), Princeton Univ. Press, Princeton, N.J., 1970, 41–59.  Google Scholar

[5]

M. Hochman, Lectures on fractal geometry and dynamics., Unpublished lecture notes, available online at http://math.huji.ac.il/ mhochman/courses/fractals-2012/course-notes.june-26.pdf. Google Scholar

[6]

P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, Ann. of Math., 189 (2019), 319-391.  doi: 10.4007/annals.2019.189.2.1.  Google Scholar

[7]

M. Wu, A proof of Furstenberg's conjecture on the intersections of $\times p$- and $\times q$-invariant sets, Ann. of Math., 189 (2019), 707-751.  doi: 10.4007/annals.2019.189.3.2.  Google Scholar

[8]

H. Yu, Bernoulli decomposition and arithmetical independence between sequences, Ergodic Theory Dynam. Systems, 41, (2021), 1560–1600. doi: 10.1017/etds.2019.117.  Google Scholar

show all references

References:
[1]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in Ergodic Theory and Topological Dynamics., Book draft, available online at https://tbward0.wixsite.com/books/entropy. Google Scholar

[2]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014.  Google Scholar

[3]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[4]

H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, in Problems in Analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969), Princeton Univ. Press, Princeton, N.J., 1970, 41–59.  Google Scholar

[5]

M. Hochman, Lectures on fractal geometry and dynamics., Unpublished lecture notes, available online at http://math.huji.ac.il/ mhochman/courses/fractals-2012/course-notes.june-26.pdf. Google Scholar

[6]

P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, Ann. of Math., 189 (2019), 319-391.  doi: 10.4007/annals.2019.189.2.1.  Google Scholar

[7]

M. Wu, A proof of Furstenberg's conjecture on the intersections of $\times p$- and $\times q$-invariant sets, Ann. of Math., 189 (2019), 707-751.  doi: 10.4007/annals.2019.189.3.2.  Google Scholar

[8]

H. Yu, Bernoulli decomposition and arithmetical independence between sequences, Ergodic Theory Dynam. Systems, 41, (2021), 1560–1600. doi: 10.1017/etds.2019.117.  Google Scholar

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