Additive Combinatorics and Ergodic Methods in Fractals

Winter/Spring 2016, University of Bristol

Type: Graduate course
Lecturer: Tuomas Sahlsten
Place: Howard House (Queen's Avenue), 4th Floor Seminar Room, at 11 am - 1 pm
Teaching: 20 hours, 2 hours per week.


Self-similarity is a notion where a set or a measure is roughly similar to small parts (components) of itself. Natural examples with this feature include self-similar sets, which are toy models for the study of irregular attractors to many hyperbolic dynamical systems. In this course we will discuss some recent breakthroughs on the theory of self-similar measures, where elements from additive combinatorics and ergodic theory are crucial. The methods presented here are actively studied in the field and so the course will present potential opportunities for new research directions.

The focus is on Hochman’s inverse theorem for entropy, which has roots in Freiman’s theorem and Bourgain’s sum-product theory. The aim is to discuss how Hochman’s work leads to the solution of Furstenberg’s projection problem of the 1-dimensional self-similar Sierpinski gasket, improvements on the Erdős problem on Bernoulli convolutions and Sinai’s problem on iterated function systems contracting on average. If time allows, we will also model the dynamics of taking component measures in the proof of Hochman’s inverse theorem with Furstenberg’s CP chains (scenery flow), which employs tools from ergodic theory to approach various other arithmetic and geometric problems.


  • Basics on self-similar sets and measures
  • Short overview of additive combinatorics
  • Entropy and its use in the study of uniformity
  • Additive combinatorics analogues for entropy
  • Multiscale analysis and convolution powers (Berry-Esseen theorem)
  • Growth of entropy of convolution powers (Kaimanovich-Vershik lemma)
  • Hochman’s inverse theorem
  • Components of self-similar measures with overlaps
  • Applications of Hochman's theorem
  • Ergodic theory of CP chains


There are not that many preliminaries required, mostly basic measure theory. Moreover, previous background on additive combinatorics and/or ergodic theory can be helpful but not necessary. The course aims to be as self-contained as possible.


Most of the material is based on:
  • M. Hochman: On self-similar sets with overlaps and inverse theorems for entropy. Annals of Mathematics 180 (2014), no. 2, pp. 773–822
  • M. Hochman: Self similar sets, entropy and additive combinatorics. Geometry and Analysis of Fractals, Springer Proceedings in Mathematics & Statistics Volume 88, 2014, pp. 225-252
Moreover, we will also explain briefly background and related topics from:
  • J. Bourgain: The discretized sum-product and projection theorems. Journal d'Analyse Mathématique 112 (2010), no. 1, pp. 193-236
  • H. Furstenberg: Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam. Systems 28 (2008), no. 2, pp.405-422
  • P. Shmerkin: On the exceptional set for absolute continuity of Bernoulli convolutions. Geometric and Functional Analysis 24 (2014), no. 3, pp. 946-958
Helpful textbooks for some background:
  • K. Falconer: Fractal Geometry, John Wiley & Sons, Ltd, 2003.
  • P. Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995
  • T. Tao, V. Vu: Additive Combinatorics, Cambridge University Press, 2006


Pass/fail: One can either write a short essay or give a presentation on a mutually agreed topic related to the course


Lecture notes: The page numbering below follows the lecture notes:
  • 1.2.2016: Self-similar sets, Hochman's theorem (pages 3-8)
  • 8.2.2016: Additive combinatorics, Freiman's theorem, multiscale analysis (pages 8-13)
  • 15.2.2016: No lecture. Overlapping a conference related to the course at ICERM: Ergodic, Algebraic and Combinatorial Methods in Dimension Theory.
  • 22.2.2016: Sum-set structure and box dimension of self-similar sets (pages 13-17)
  • 29.2.2016: Heuristic proof of Hochman's theorem, convolution (pages 17-23)
  • 7.3.2016: Entropy, Tao's inverse theorem, Hochman's inverse theorem (pages 23-28)
  • 14.3.2016: Entropy from component measures, proof of Hochman's theorem for self-similar measures (pages 29-35)
  • 21.3.2016: No lecture. Easter vacation + BMC 2016
  • 28.3.2016: No lecture. Easter vacation
  • 4.4.2016: No lecture. Easter vacation
  • 11.4.2016: No lecture. A number of people attending the lectures are away.
  • 18.4.2016: Proof of the inverse theorem for the entropy, part I: Plünnecke-Ruzsa inequality, Kaimanovich-Vershik lemma, Berry-Esseen theorem (pages 35-40)
  • 25.4.2016: Proof of the inverse theorem for the entropy, part II: Components of large self-convolutions, Applying Berry-Esseen and Kaimanovich-Vershik, Completion of the proof (pages 40-47)
  • 4.5.2016: Wrapping up the course. Reserved for talks by the students. Four talks will be held during 11:30 am - 2:30 pm at 4th floor seminar room. We will have lunch together during the talks.