Contents
Preface ix
Acknowledgments x
Chapter 1. Higher order Fourier analysis 1
§1.1. Equidistribution of polynomial sequences in tori 2
§1.2. Roth’s theorem 26
§1.3. Linear patterns 45
§1.4. Equidistribution of polynomials over finite fields 59
§1.5. The inverse conjecture for the Gowers norm I. The finite field
case 74
§1.6. The inverse conjecture for the Gowers norm II. The integer
case 92
§1.7. Linear equations in primes 109
Chapter 2. Related articles 129
§2.1. Ultralimit analysis and quantitative algebraic geometry 130
§2.2. Higher order Hilbert spaces 149
§2.3. The uncertainty principle 162
Bibliography 179
Index 185
vii
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