Additive Combinatorics and Ergodic Methods in Fractals

Spring 2017, University of Manchester

Type: Graduate course
Lecturer: Tuomas Sahlsten
Place: Alan Turing Building, Frank Adams 1 classroom on Thursdays at 10 am - 12 noon
Teaching: 22 hrs of lectures over 11 weeks (+ up to 11 hrs of tutorial sessions)

Overview

This course is an introduction to the sum-product theory in additive combinatorics by presenting the recently developed measure theoretical analogues by Kaimanovich, Vershik, Tao and Hochman. The theory has wide range of applications in the theory of random walks on groups, ergodic theory, geometric measure theory and fractal geometry. The methods presented here are actively studied in the field and so the course will present potential opportunities for new research directions.

We will concentrate on Hochman’s inverse theorem for the entropy of convolutions (Annals of Maths. 180 (2014), no. 2, pp. 773–822) and 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 permits, we will also discuss algebraic applications and connections (Bourgain-Gamburd theory).

Topics

  • Short overview of additive combinatorics
  • Entropy and its use in the study of uniformity
  • Additive combinatorics analogues for entropy (convolutions)
  • Multiscale analysis and convolution powers (Berry-Esseen theorem)
  • Growth of entropy of convolution powers (Kaimanovich-Vershik lemma)
  • Hochman’s inverse theorem
  • Applications of Hochman's inverse theorem
  • Random walks on groups (Bourgain-Gamburd theory) and related topics

Prerequisites

Mostly the understanding of basic measure theory is required. Moreover, knowledge of ergodic theory can be helpful but not necessary. Previous background on additive combinatorics may help in understanding the topic.

Bibliography

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

Assessment

2 assignments

Lecture notes

  • LaTeX notes (pdf) of the lectures. (Dated 2. March, 2017)

Schedule

  • 2.2.2017: Freiman's theorem and Bourgain's inverse theorem (Pages 4-7)
  • 9.2.2017: Hochman's inverse theorem (Pages 7-10)
  • 16.2.2017: Convolution and entropy (Pages 11-14)
  • 23.2.2017: Tao's inverse theorem for entropy, multiscale analysis and Hochman's inverse theorem for entropy (Pages 15-16)
  • 2.3.2017: Multiscale analysis (continued), ideas of the proof of Hochman's theorem: how Berry-Esseen comes up and Kaimanovich-Vershik lemma (Covers parts from Pages 16-31)
  • 9.3.2017: Multiscale analysis notations, local entropy averages (Pages 18 and 25)
  • 16.3.2017: Applying Berry-Esseen theorem (Pages 23-27)
  • 23.3.2017: Proof of Kaimanovich-Vershik lemma and completition of the proof (Pages 28-31)
  • 30.3.2017: Applications
  • Easter break
  • 27.4.2017: Applications
  • 4.5.2017: Applications