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
v
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