|
Preface | ii |
|
Overview | ix |
|
Notation Guide | xv |
|
|
Part 1. Additive Number Systems | 1 |
|
Chapter 1. Background From Analysis | 3 |
|
1.1 Series and infinite products | 4 |
|
1.2 Power series expansions | 9 |
|
1.3 Big O, little o, and $\sim$ notation | 12 |
|
1.4 The radius of convergence and $RT_\rho$ | 13 |
|
1.5 Growth rate of coefficients | 15 |
|
Chapter 2. Counting Functions, Fundamental Identities | 17 |
|
2.1 Defining additive number systems | 17 |
|
2.2 Examples of additive number systems | 21 |
|
2.3 Counting functions, fundamental identities | 25 |
|
2.4 Global counts | 33 |
|
2.5 Alternate version of the fundamental identity | 34 |
|
2.6 Reduced additive number systems | 37 |
|
2.7 Finitely generated number systems | 39 |
|
2.8 $a^\star (n)$ is eventually positive | 43 |
|
Chapter 3. Density and Partition Sets | 45 |
|
3.1 Asymptotic density | 45 |
|
3.2 Dirichlet density | 48 |
|
3.3 The standard assumption | 51 |
|
3.4 The set of additives of an element | 51 |
|
3.5 Partition sets | 54 |
|
3.6 Generating series of partition sets | 57 |
|
3.7 Partition sets have Dirichlet density | 58 |
|
3.8 Schur's Tauberian Theorem | 62 |
|
3.9 Simple partition sets | 67 |
|
3.10 The asymptotic density of $\gamma P$ | 68 |
|
3.11 Adding an indecomposable | 70 |
|
Chapter 4. The Case $\rho = 1$ | 75 |
|
4.1 The fundamental results when $\rho = 1$ | 75 |
|
4.2 The Bateman and Erd\"os Result | 77 |
|
4.3 Stewart's Sum Theorem | 83 |
|
4.4 $\rho = 1$ does not imply $RT_1$ | 84 |
|
4.5 Further Results on $RT_1$ | 85 |
|
Chapter 5. The Case $0< \rho < 1$ | 87 |
|
5.1 Compton's Tauberian Theorem | 87 |
|
5.2 When is $\delta (B) = 0$? | 90 |
|
5.3 The density of $\overline{B}$ | 91 |
|
5.4 Compton's Density Theorem | 92 |
|
5.5 The Knopfmacher, Knopfmacher, Warlimont asymptotics | 93 |
|
|
|
Chapter 6. Monadic Second-Order Limit Laws | 103 |
|
6.1 First-order logic | 103 |
|
6.2 Monadic second-order logic | 104 |
|
6.3 Asymptotic density of subsets of $K$ | 106 |
|
6.4 Quantifier rank and equivalent formulas | 107 |
|
6.5 Ehrenfeucht-Fraiss\'e games | 109 |
|
6.6 Adequate classes of structures | 119 |
|
|
Part 2. Multiplicative Number Systems | 125 |
|
Chapter 7. Background from Analysis | 127 |
|
7.1 Series and infinite products | 127 |
|
7.2 Dirichlet series expansions | 133 |
|
7.3 The abcissa of convergence and $RV_\alpha $ | 136 |
|
7.4 Growth rate of coefficients | 140 |
|
Chapter 8. Counting Functions and Fundamental Identities | 143 |
|
8.1 Defining multiplicative number systems | 143 |
|
8.2 Examples of multiplicative number systems | 144 |
|
8.3 Counting functions, fundamental identities | 145 |
|
8.4 Alternate version of the fundamental identity | 153 |
|
8.5 Finitely generated multiplicative number systems | 156 |
|
Chapter 9. Density and Partition Sets | 159 |
|
9.1 Asymptotic density | 159 |
|
9.2 Dirichlet density | 160 |
|
9.3 The standard assumption | 164 |
|
9.4 The set of multiples of an element | 164 |
|
9.5 Partition sets | 166 |
|
9.6 Generating series of partition sets | 169 |
|
9.7 Partition sets have Dirichlet density | 170 |
|
9.8 Discrete multiplicative number systems | 174 |
|
9.9 When sets $bA$ have global asymptotic density | 177 |
|
9.10 The strictly multiplicative case and $RV_\alpha$ | 181 |
|
9.11 The discrete case and $RV_\alpha$ | 181 |
|
9.12 Analog of Schur's Tauberian Theorem | 182 |
|
9.13 Simple partition sets | 188 |
|
9.14 The asymptotic density of $P^\gamma$ | 189 |
|
9.15 Adding an indecomposable | 191 |
|
9.16 First conjecture | 194 |
|
Chapter 10. The Case $\alpha = 0$ | 195 |
|
10.1 The fundamental results when $\alpha = 0$ | 195 |
|
10.2 Odlyzko's Product Theorem | 196 |
|
10.3 Further results on $RV_0$ | 198 |
|
10.4 Second conjecture | 199 |
|
Chapter 11. The Case $0 < \alpha < \infty $ | 201 |
|
11.1 Analog of Compton's Tauberian Theorem | 201 |
|
11.2 When is $\Delta (B) = 0$? | 204 |
|
11.3 The density of $\overline{B}$ | 205 |
|
11.4 S\'ark\"ozy's Density Theorem | 206 |
|
11.5 Generalizing Oppenheim's asymptotics | 208 |
|
11.6 Third conjecture | 216 |
|
Chapter 12. First-Order Limit Laws | 217 |
|
12.1 Asymptotic density of subsets of $K$ | 217 |
|
12.2 The Feferman-Vaught Theorem | 218 |
|
12.3 Skolem's analysis of the first-order calculus of classes | 223 |
|
12.4 $K_\phi$ is a disjoint union of partition classes | 226 |
|
12.5 Applications | 227 |
|
12.6 Finite dimensional structures | 231 |
|
12.7 The main problem | 231 |
|
Appendix A. Formal Power Series | 233 |
|
|
|
Appendix B. Refined Counting | 251 |
|
|
Appendix C. Consequences of $\delta (P) = 0$ | 261 |
|
|
|
Appendix D. On the Monotonicity of $a(n)$ When
$p(n) \leq 1$ | 269 |
|
|
Appendix E. Results of Woods | 273 |
|
|
Bibliography | 281 |
|
Symbol Index | 285 |
|
Subject Index | 287 |
|
|