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.
Overview
Selfsimilarity is a notion where a set or a measure is roughly similar to small parts (components) of itself. Natural examples with this feature include selfsimilar 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 selfsimilar 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 sumproduct theory. The aim is
to discuss how Hochman’s work leads to the solution of Furstenberg’s
projection problem of the 1dimensional selfsimilar 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.
Topics
 Basics on selfsimilar 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 (BerryEsseen theorem)
 Growth of entropy of convolution powers (KaimanovichVershik lemma)
 Hochman’s inverse theorem
 Components of selfsimilar measures with overlaps
 Applications of Hochman's theorem
 Ergodic theory of CP chains
Prerequisites
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 selfcontained as possible.
Bibliography
Most of the material is based on:
 M. Hochman: On selfsimilar 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.
225252
Moreover, we will also explain briefly background and related topics
from:
 J. Bourgain: The discretized sumproduct and projection
theorems. Journal d'Analyse Mathématique 112 (2010), no. 1,
pp. 193236
 H. Furstenberg: Ergodic fractal measures and dimension
conservation. Ergodic Theory Dynam. Systems 28 (2008), no. 2, pp.405422
 P. Shmerkin: On the exceptional set for absolute continuity of
Bernoulli convolutions. Geometric and Functional Analysis 24
(2014), no. 3, pp. 946958
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
Grading
Pass/fail: One can either write a short essay or give a presentation on a mutually agreed topic related
to the course
Schedule
Lecture notes:
The page numbering below follows the lecture notes:

1.2.2016: Selfsimilar sets, Hochman's theorem (pages 38)

8.2.2016: Additive combinatorics, Freiman's theorem, multiscale
analysis (pages 813)

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: Sumset structure and box dimension of selfsimilar sets (pages 1317)

29.2.2016: Heuristic proof of Hochman's theorem, convolution (pages 1723)

7.3.2016: Entropy, Tao's inverse theorem, Hochman's inverse theorem (pages 2328)

14.3.2016: Entropy from component measures, proof of Hochman's
theorem for selfsimilar measures (pages 2935)

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:
PlünneckeRuzsa inequality, KaimanovichVershik lemma,
BerryEsseen theorem (pages 3540)

25.4.2016: Proof of the inverse theorem for the entropy, part II:
Components of large selfconvolutions, Applying BerryEsseen and KaimanovichVershik, Completion of the proof (pages 4047)

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.