Contents

Preface vii

A remark on notation viii

Acknowledgments ix

Chapter 1. Expository Articles 1

§1.1. The blue-eyed islanders puzzle 1

§1.2. Kleiner’s proof of Gromov’s theorem 2

§1.3. The van der Corput lemma, and equidistribution on

nilmanifolds 9

§1.4. The strong law of large numbers 15

§1.5. Tate’s proof of the functional equation 22

§1.6. The divisor bound 31

§1.7. The Lucas-Lehmer test for Mersenne primes 36

§1.8. Finite subsets of groups with no finite models 41

§1.9. Small samples, and the margin of error 47

§1.10. Non-measurable sets via non-standard analysis 56

§1.11. A counterexample to a strong polynomial Freiman-Ruzsa

conjecture 58

§1.12. Some notes on “non-classical” polynomials in finite

characteristic 61

§1.13. Cohomology for dynamical systems 67

Chapter 2. Ergodic Theory 75

§2.1. Overview 75

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