Contents

Preface	xi
Notation	xiii
Chapter 1. Uniform Distribution	1

1.	Qualitative theory	1
2.	Quantitative relations	3
3.	Trigonometric approximation	10
4.	Notes	13
	References	15

Chapter 2. van der Corput Sets

17

1.	Introduction	17
2.	Extremal measures	23
3.	Relations between alpha, beta_infinity, beta_2	25
4.	Corollaries	28
5.	A sufficient condition	31
6.	Intersective sets	34
7.	Heilbronn sets	35
8.	Notes	37
	References	37

Chapter 3. Exponential Sums I:
The Methods of Weyl and van der Corput	39

1.	Introduction	39
2.	Weyl's method	39
3.	van der Corput's method	46
4.	Exponent pairs	56
5.	Notes	60
	References	61

Chapter 4. Exponential Sums II:
Vinogradov's Method	65

1.	Introduction	65
2.	Vinogradov's Mean Value Theorem	69
3.	A bound for Weyl sums	76
4.	An alternative derivation	79
5.	Notes	81
	References	82

Chapter 5. An Introduction to Turán's Method

85

1.	Introduction	85
2.	Turán's First Main Theorem	86
3.	Fabry's Gap Theorem	89
4.	Longer ranges of nu	91
5.	Turán's Second Main Theorem	93
6.	Special coefficients bn	97
7.	Notes	102
	References	105

Chapter 6. Irregularities of Distribution

109

1.	Introduction	109
2.	Squares	110
3.	Disks	111
4.	Decay of the Fourier Transform	114
5.	Families allowing translation, scaling and rotation	119
6.	Notes	120
	References	122

Chapter 7. Mean and Large Values
of Dirichlet Polynomials	125

1.	Introduction	125
2.	Mean values via trigonometric approximation	127
3.	Majorant principles	131
4.	Review of Elementary Operator Theory	134
5.	Mean values via Hilbert's inequality	137
6.	Large value estimates	140
7.	Notes	143
	References	146

Chapter 8. Distribution of Reduced
Residue Classes in Short Intervals	151

1.	Introduction	151
2.	A probabilistic model	153
3.	An approach by Fourier techniques	154
4.	The fundamental lemma	156
5.	Notes	160
	References	161

Chapter 9. Zeros of L-functions

163

1.	Introduction	163
2.	Least Character Non-Residues	164
3.	Clumps of zeros	168
4.	The Deuring-Heilbronn phenomenon	172
5.	Notes	176
	References	177

Chapter 10. Small Polynomials
with Integral Coefficients	179

1.	Introduction	179
2.	The Gorskov-Wirsing Polynomials	183
3.	Notes	188
	References	190

Appendix: Some Unsolved Problems

195

1.	Uniform Distribution	195
2.	van der Corput Sets	196
3.	Weyl Sums	196
4.	van der Corput's Method	197
5.	Turán's Method	197
6.	Irregularities of Distribution	198
7.	Mean and Large Values of Dirichlet Polynomials	198
8.	Reduced Residues in Short Intervals	200
9.	Zeros of L-Functions	201
10.	Small Polynomials with Integral Coefficients	201
11.	Character Sums	202
12.	Diophantine Approximation	202
13.	Metric Diophantine Approximation	204
14.	Algebraic Integers	205
15.	Trigonometric Polynomials	206
16.	Miscellaneous	207
	References	210

Index

215