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From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath (Brin Prize article)

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  • M. Hochman's work on the dimension of self-similar sets has given impetus to resolving other questions regarding fractal dimension. We describe Hochman's approach and its influence on the subsequent resolution by P. Shmerkin of the conjecture on the dimension of the intersection of $ \times p $- and $ \times q $-Cantor sets.

    Mathematics Subject Classification: Primary: 28A80, 37C45; Secondary: 11K55, 11B30.

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  • [1] H. Furstenberg, Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis (Symposium in honor of Salomon Bochner, Princeton University Press, Princeton, N.J. 1969), 41–59, 1970.
    [2] 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.
    [3] M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773-822.  doi: 10.4007/annals.2014.180.2.7.
    [4] P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$-norms of convolutions, Ann. of Math. (2), 189 (2019), 319–391. doi: 10.4007/annals.2019.189.2.1.
    [5] M. Wu, A proof of Furstenberg's conjecture on the intersection of $\times p$ and $\times q$-invariant sets, arXiv: 1609.08053v3, February 2019.
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