The book has five chapters,[3] each largely self-contained[4][5] and loosely organized around different techniques used to solve problems in this area,[4] with an appendix on the background material in number theory needed for reading the book.[3] Rather than being concerned with specific sequences such as the prime numbers or square numbers, its topic is the mathematical theory of sequences in general.[6][7]
The first chapter considers the natural density of sequences, and related concepts such as the Schnirelmann density. It proves theorems on the density of sumsets of sequences, including Mann's theorem that the Schnirelmann density of a sumset is at least the sum of the Schnirelmann densities and Kneser's theorem on the structure of sequences whose lower asymptotic density is subadditive. It studies essential components, sequences that when added to another sequence of Schnirelmann density between zero and one, increase their density, proves that additive bases are essential components, and gives examples of essential components that are not additive bases.[3][6][7][8]
The second chapter concerns the number of representations of the integers as sums of a given number of elements from a given sequence, and includes the Erdős–Fuchs theorem according to which this number of representations cannot be close to a linear function. The third chapter continues the study of numbers of representations, using the probabilistic method; it includes the theorem that there exists an additive basis of order two whose number of representations is logarithmic, later strengthened to all orders in the Erdős–Tetali theorem.[3][6][7][8]
After a chapter on sieve theory and the large sieve (unfortunately missing significant developments that happened soon after the book's publication),[6][7] the final chapter concerns primitive sequences of integers, sequences like the prime numbers in which no element is divisible by another. It includes Behrend's theorem that such a sequence must have logarithmic density zero, and the seemingly-contradictory construction by Abram Samoilovitch Besicovitch of primitive sequences with natural density close to 1/2. It also discusses the sequences that contain all integer multiples of their members, the Davenport–Erdős theorem according to which the lower natural and logarithmic density exist and are equal for such sequences, and a related construction of Besicovitch of a sequence of multiples that has no natural density.[3][6][7]
