Fractals in probability and Analysis

This book provides a broad introduction to the study of fractals that arise naturally in probability and analysis. As noted in the review by David Croydon, it starts by setting out the classical notions of dimension (i.e., Minkowski, Hausdorff, packing) and the key basic techniques applied in their study, such as the mass distribution principle, and Frostman’s theory and capacity. It then proceeds to introduce some of the well-known, central examples of fractals, namely self-affine sets, the Weierstrass nowhere differentiable function, and Brownian motion. Actually, the two introductory chapters on Brownian motion are relatively extensive, incorporating not only basic definitions and properties, such as nowhere differentiability and dimension, but also the deep links between Brownian motion and potential theory, as well as conformal invariance. Following this, the book goes on to cover more novel aspects of the subject, including the relationship between capacity and the hitting probabilities of discrete Markov processes, a discussion concerning Besicovitch-Kakeya sets, and a presentation of Jones’ Travelling Salesman Theorem.
The material is at an appropriate level for a graduate (or possibly advanced undergraduate) course. The focus is on exposition of the main ideas, rather than the most advanced statements of results. Nonetheless, through this accessible approach, the book touches on several avenues of active research. ( Moreover, all the main results are illustrated with numerous examples, and the text includes several hundred exercises at a range of difficulties, together with hints and solutions for a number of these.

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Related open problems

If you have ideas or comments pertaining to the open problems listed below, please send email to


Let K be the four corner set, which is the Cartesian square of the middle-half Cantor set. Consider stage n in the construction of K, which is a union K_n of 4^n squares of sidelength 1/4^n. Let F_n be the Favard length of K_n, which is the avarage length of projections of K_n on lines. How fast does F_n tend to zero? The best upper bound known is in Nazarov, Fedor, Yuval Peres, and Alexander Volberg. “The power law for the Buffon needle probability of the four-corner Cantor set.” St. Petersburg Mathematical Journal 22, no. 1 (2011): 61-72.  (Link to PDF) and the best lower bound is in Bateman, Michael, and Alexander Volberg. “An estimate from below for the Buffon needle probability ot the four-corner Cantor set.” Mathematical research letters 17, no. 5 (2010): 959-968.


What is the minimal Hausdorff dimension of a Kakeya set in three-dimensional space (i.e., a compact set containing line segments in all directions)? See Katz, Nets, and Terence Tao. “Recent progress on the Kakeya conjecture.” Publicacions matemàtiques (2002): 0161-179.

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