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A new dynamical proof of the Shmerkin–Wu theorem

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  • 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.

    Mathematics Subject Classification: Primary: 11K55, 37A45; Secondary: 28A50, 28A80, 37C45.


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  • [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.
    [2] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd edition, John Wiley & Sons, Ltd., Chichester, 2014.
    [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.
    [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.
    [5] M. Hochman, Lectures on fractal geometry and dynamics., Unpublished lecture notes, available online at http://math.huji.ac.il/ mhochman/courses/fractals-2012/course-notes.june-26.pdf.
    [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.
    [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.
    [8] H. Yu, Bernoulli decomposition and arithmetical independence between sequences, Ergodic Theory Dynam. Systems, 41, (2021), 1560–1600. doi: 10.1017/etds.2019.117.
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