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 |