Sequences

I. Addition of Sequences: Study of Density Relationships.- 1. Introduction and notation.- 2. Schnirelmann density and Schnirelmann's theorems. Besicovitch's theorem.- 3. Essential components and complementary sequences.- 4. The theorems of Mann, Dyson, and van der Corput.- 5. Bases and non-basic essential components.- 6. Asymptotic analogues and p-adic analogues.- 7. Kneser's theorem.- 8. Kneser's theorem (continued): the ?-transformations.- 9. Kneser's theorem (continued): proof of Theorem 19-sequence functions associated with the derivations of a system.- 10. Kneser's theorem (continued): proofs of Theorems 16? and 17?.- 11. Hanani's conjecture.- II. Addition of Sequences: Study of Representation Functions by Number Theoretic Methods.- 1. Introduction.- 2. Auxiliary results from the theory of finite fields.- 3. Sidon's problems.- 4. The Erdös-Fuchs theorem.- III. Addition of Sequences: Study of Representation Functions by Probability Methods.- 1. Introduction.- 2. Principal results.- 3. Finite probability spaces: informal discussion.- 4. Measure theory: basic definitions.- 5. Measure theory: measures on product spaces.- 6. Measure theory: simple functions.- 7. Probability theory: basic definitions and terminology.- 8. Auxiliary lemmas.- 9. Probability theory: some fundamental theorems.- 10. Probability measures on the space of (positive) integer sequences.- 11. Preparation for the proofs of Theorems 1-4.- 12. Proof of Theorem 1.- 13. Proof of Theorem 2.- 14. Proof of Theorem 3.- 15. Quasi-independence of the variables rn.- 16. Proof of Theorem 4-sequences of pseudo-squares.- IV. Sieve Methods.- 1. Introduction.- 2. Notation andpreliminaries.- 3. The number of natural numbers not exceeding x not divisible by any prime less than y.- 4. The generalized sieve problem.- 5. The Viggo Brun method.- 6. Selberg's upper-bound method: informal discussion.- 7. Selberg's upper-bound method.- 8. Selberg's lower-bound method.- 9. Selberg's lower-bound method: further