''Sequences'' is a mathematical monograph on

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. Fo ...

s. It was written by

Heini Halberstam Heini Halberstam (11 September 1926 oreen Halberstam, wife/ref> – 25 January 2014) was a Czech-born British mathematician, working in the field of analytic number theory. He is remembered in part for the Elliott–Halberstam conjecture from 1 ...

and

Klaus Roth Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De ...

, published in 1966 by the

Clarendon Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...

, and republished in 1983 with minor corrections by

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

. Although planned to be part of a two-volume set, the second volume was never published.

Topics

The book has five chapters, each largely self-contained and loosely organized around different techniques used to solve problems in this area, with an appendix on the background material in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

needed for reading the book. Rather than being concerned with specific sequences such as the

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

s or

square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...

s, its topic is the mathematical theory of sequences in general. The first chapter considers the

natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the ...

of sequences, and related concepts such as the

Schnirelmann density In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.Schnirelmann, L.G. (1930).On the a ...

. It proves theorems on the density of

sumset In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is, :A + B = \. The n-fo ...

s 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. 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 In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, stating that the average order of this number ...

according to which this number of representations cannot be close to a

linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...

. The third chapter continues the study of numbers of representations, using the

probabilistic method The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects fr ...

; 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 In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive bases of every order. More specifically, it states that for every fixed integer h \geq 2, there exists a subset o ...

. After a chapter on

sieve theory Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed lim ...

and the

large sieve The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of sieve where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein o ...

(unfortunately missing significant developments that happened soon after the book's publication), the final chapter concerns primitive sequences of integers, sequences like the

s 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 Abram Samoilovitch Besicovitch (or Besikovitch) (russian: link=no, Абра́м Само́йлович Безико́вич; 23 January 1891 – 2 November 1970) was a Russian mathematician, who worked mainly in England. He was born in Berdyansk ...

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 In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent. Let A=a_1,a_2,\dots be a sequence of positive integers. Then the multiples of A are another set M ...

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.

Audience and reception

This book is aimed at other mathematicians and students of mathematics; it is not suitable for a general audience. However, reviewer

J. W. S. Cassels John William Scott "Ian" Cassels, FRS (11 July 1922 – 27 July 2015) was a British mathematician. Biography Cassels was educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh. He went on to study a ...

suggests that it could be accessible to advanced undergraduates in mathematics. Reviewer E. M. Wright notes the book's "accurate scholarship", "most readable exposition", and "fascinating topics". Reviewer Marvin Knopp describes the book as "masterly", and as the first book to overview additive combinatorics. Similarly, although Cassels notes the existence of material on additive combinatorics in the books ''Additive Zahlentheorie'' (Ostmann, 1956) and ''Addition Theorems'' (Mann, 1965), he calls this "the first connected account" of the area, and reviewer

Harold Stark Harold Mead Stark (born August 6, 1939 in Los Angeles, California) is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the ear ...

notes that much of material covered by the book is "unique in book form". Knopp also praises the book for, in many cases, correcting errors or deficiencies in the original sources that it surveys. Reviewer

writes that the book "should be a standard reference in this area for years to come".

References

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Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also ...

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zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...

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Journal of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematica ...

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Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

, pages=943–957, title=Review of ''Sequences'', url=https://projecteuclid.org/euclid.bams/1183533164, volume=77, year=1971, doi=10.1090/s0002-9904-1971-12812-4, doi-access=free Additive combinatorics Integer sequences Mathematics books 1966 non-fiction books 1983 non-fiction books Clarendon Press books