Pablo Shmerkin
I am a Professor at the University of British Columbia (on leave from T. Di Tella University). My pronouns are he/him.
My work generally spans the areas of fractal geometry, ergodic theory and analysis, although I am also interested in connections to probability and combinatorics.
In particular, my recent work is concerned with:
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Geometric properties of (random and deterministic) fractals of dynamical, arithmetic and combinatorial origin. In particular, geometrical aspects of dynamical rigidity and smoothness of self-similar measures.
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Combinatorial problems in fractal geometry (distance sets, direction sets, Kakeya-type sets, projection theorems, etc).
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Applications of fractal geometry in ergodic theory and analysis.
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Self-affine sets and the subadditive thermodynamic formalism.
RECENT ARTICLES
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Tuomas Orponen, Pablo Shmerkin and Hong Wang. Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem. https://arxiv.org/abs/2209.00348.
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Pablo Shmerkin and Hong Wang. On the distance sets spanned by sets of dimension d/2 in R^d. https://arxiv.org/abs/2112.09044 .
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Amir Algom, Simon Baker and Pablo Shmerkin. On normal numbers and self-similar measures. https://arxiv.org/abs/2111.10082 . Adv. Math. 399 (2022), Paper No. 108276, 17 pp.
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Pablo Shmerkin. Slices and distances: on two problems of Furstenberg and Falconer. https://arxiv.org/abs/2109.12157 . This is my contribution to the ICM 2022 Proceedings and provides a high level overview of recent progress around Furstenberg's slicing conjecture and the Falconer distance set problem.
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Pablo Shmerkin and Ville Suomala. New bounds on Cantor maximal operators. https://arxiv.org/abs/2106.14818 .Rev. Un. Mat. Argentina 64 (2022), no. 1, 69--86. (Special issue dedicated to the Mathematical Congress of the Americas 2021.)
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Tuomas Orponen and Pablo Shmerkin. On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane. https://arxiv.org/abs/2106.03338 . To appear in Duke Math. J.
UPCOMING EVENTS
Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory