Klaus Roth
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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 In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half ...
on the
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...
of algebraic numbers. He was also a winner of the
De Morgan Medal The De Morgan Medal is a prize for outstanding contribution to mathematics, awarded by the London Mathematical Society. The Society's most prestigious award, it is given in memory of Augustus De Morgan, who was the first President of the society ...
and the
Sylvester Medal The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry a ...
, and a Fellow of the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
. Roth moved to England as a child in 1933 to escape the Nazis, and was educated at the
University of Cambridge The University of Cambridge is a public collegiate research university in Cambridge, England. Founded in 1209 and granted a royal charter by Henry III in 1231, Cambridge is the world's third oldest surviving university and one of its most pr ...
and
University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
, finishing his doctorate in 1950. He taught at University College London until 1966, when he took a chair at
Imperial College London Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cu ...
. He retired in 1988. Beyond his work on Diophantine approximation, Roth made major contributions to the theory of progression-free sets in
arithmetic combinatorics In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Scope Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (ad ...
and to the theory of irregularities of distribution. He was also known for his research on sums of powers, on the large sieve, on the
Heilbronn triangle problem In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed ...
, and on
square packing in a square Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side . If is an integer, the answer is , but the precise, or even asymptotic, amount ...
. He was a coauthor of the book ''
Sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
'' 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. For ...
s.


Biography


Early life

Roth was born to a Jewish family in Breslau,
Prussia Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
, on 29 October 1925. His parents settled with him in London to escape Nazi persecution in 1933, and he was raised and educated in the UK. His father, a solicitor, had been exposed to poison gas during
World War I World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
and died while Roth was still young. Roth became a pupil at
St Paul's School, London (''By Faith and By Learning'') , established = , closed = , type = Independent school Public school , religion = Church of England , president = , h ...
from 1939 to 1943, and with the rest of the school he was evacuated from London to
Easthampstead Park Easthampstead Park is a Victorian mansion in the civil parish of Bracknell in the English county of Berkshire. It is now a conference centre. Location Since the demise of Easthampstead parish, the house has been located in the western extreme ...
during
The Blitz The Blitz was a German bombing campaign against the United Kingdom in 1940 and 1941, during the Second World War. The term was first used by the British press and originated from the term , the German word meaning 'lightning war'. The Germa ...
. At school, he was known for his ability in both chess and mathematics. He tried to join the
Air Training Corps The Air Training Corps (ATC) is a British volunteer-military youth organisation. They are sponsored by the Ministry of Defence and the Royal Air Force. The majority of staff are volunteers, and some are paid for full-time work – including C ...
, but was blocked for some years for being German and then after that for lacking the coordination needed for a pilot.


Mathematical education

Roth read mathematics at Peterhouse, Cambridge, and played first board for the Cambridge chess team, finishing in 1945. Despite his skill in mathematics, he achieved only
third-class honours The British undergraduate degree classification system is a grading structure for undergraduate degrees or bachelor's degrees and integrated master's degrees in the United Kingdom. The system has been applied (sometimes with significant variati ...
on the
Mathematical Tripos The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. It is the oldest Tripos examined at the University. Origin In its classical nineteenth-century form, the tripos was ...
, because of his poor test-taking ability. His Cambridge tutor, John Charles Burkill, was not supportive of Roth continuing in mathematics, recommending instead that he take "some commercial job with a statistical bias". Instead, he briefly became a schoolteacher at
Gordonstoun Gordonstoun School is a co-educational independent school for boarding and day pupils in Moray, Scotland. It is named after the estate owned by Sir Robert Gordon in the 17th century; the school now uses this estate as its campus. It is locate ...
, between finishing at Cambridge and beginning his graduate studies. On the recommendation of
Harold Davenport Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar Scho ...
, he was accepted in 1946 to a master's program in mathematics at
University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
, where he worked under the supervision of
Theodor Estermann Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born American mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean plan ...
. He completed a master's degree there in 1948, and a doctorate in 1950. His dissertation was ''Proof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power''.


Career

On receiving his master's degree in 1948, Roth became an assistant lecturer at University College London, and in 1950 he was promoted to lecturer. His most significant contributions, on Diophantine approximation, progression-free sequences, and discrepancy, were all published in the mid-1950s, and by 1958 he was given the Fields Medal, mathematicians' highest honour. However, it was not until 1961 that he was promoted to full professor. During this period, he continued to work closely with Harold Davenport. He took sabbaticals at the
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
in the mid-1950s and mid-1960s, and seriously considered migrating to the United States.
Walter Hayman Walter Kurt Hayman FRS (6 January 1926 – 1 January 2020) was a British mathematician known for contributions to complex analysis. He was a professor at Imperial College London. Life and work Hayman was born in Cologne, Germany, the son of ...
and
Patrick Linstead Sir (Reginald) Patrick Linstead CBE, DSc, HonDSc, DIC, HonFCGI, HonMIMM, FRS (28 August 1902, in London – 22 September 1966, in London) was an English chemist. Biography Patrick Linstead was born on 28 August 1902 in Southgate, London, the s ...
countered this possibility, which they saw as a threat to British mathematics, with an offer of a chair in pure mathematics at
Imperial College London Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cu ...
, and Roth accepted the chair in 1966. He retained this position until official retirement in 1988. He remained at Imperial College as Visiting Professor until 1996. Roth's lectures were usually very clear but could occasionally be erratic. The
Mathematics Genealogy Project The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. By 31 December 2021, it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a ty ...
lists him as having only two doctoral students, but one of them, William Chen, who continued Roth's work in discrepancy theory, became a Fellow of the
Australian Mathematical Society The Australian Mathematical Society (AustMS) was founded in 1956 and is the national society of the mathematics profession in Australia. One of the Society's listed purposes is to promote the cause of mathematics in the community by representing ...
and head of the mathematics department at Macquarie University.


Personal life

In 1955, Roth married Mélèk Khaïry, who had attracted his attention when she was a student in his first lecture; Khaïry was a daughter of Egyptian senator Khaïry Pacha She came to work for the psychology department at University College London, where she published research on the effects of toxins on rats. On Roth's retirement, they moved to Inverness; Roth dedicated a room of their house to Latin dancing, a shared interest of theirs. Khaïry died in 2002, and Roth died in Inverness on 10 November 2015 at the age of 90. They had no children, and Roth dedicated the bulk of his estate, over one million pounds, to two health charities "to help elderly and infirm people living in the city of Inverness". He sent the Fields Medal with a smaller bequest to Peterhouse.


Contributions

Roth was known as a problem-solver in mathematics, rather than as a theory-builder. Harold Davenport writes that the "moral in Dr Roth's work" is that "the great unsolved problems of mathematics may still yield to direct attack, however difficult and forbidding they appear to be, and however much effort has already been spent on them". His research interests spanned several topics 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, "Mat ...
,
discrepancy theory In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, name ...
, and the theory of
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. For ...
s.


Diophantine approximation

The subject of
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...
seeks accurate approximations of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s by
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s. The question of how accurately algebraic numbers could be approximated became known as the Thue–Siegel problem, after previous progress on this question by
Axel Thue Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-calle ...
and
Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
. The accuracy of approximation can be measured by the approximation exponent of a number x, defined as the largest number e such that x has infinitely many rational approximations p/q with , x-p/q, <1/q^e. If the approximation exponent is large, then x has more accurate approximations than a number whose exponent is smaller. The smallest possible approximation exponent is two: even the hardest-to-approximate numbers can be approximated with exponent two using
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s. Before Roth's work, it was believed that the algebraic numbers could have a larger approximation exponent, related to the degree of the polynomial defining the number. In 1955, Roth published what is now known as
Roth's theorem In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half ...
, completely settling this question. His theorem falsified the supposed connection between approximation exponent and degree, and proved that, in terms of the approximation exponent, the algebraic numbers are the least accurately approximated of any irrational numbers. More precisely, he proved that for irrational algebraic numbers, the approximation exponent is always exactly two. In a survey of Roth's work presented by
Harold Davenport Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory. Early life Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar Scho ...
to the International Congress of Mathematicians in 1958, when Roth was given the Fields Medal, Davenport called this result Roth's "greatest achievement".


Arithmetic combinatorics

Another result called "
Roth's theorem In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half ...
", from
1953 Events January * January 6 – The Asian Socialist Conference opens in Rangoon, Burma. * January 12 – Estonian émigrés found a government-in-exile in Oslo. * January 14 ** Marshal Josip Broz Tito is chosen President of Yug ...
, is in
arithmetic combinatorics In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Scope Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (ad ...
and concerns sequences of integers with no three in arithmetic progression. These sequences had been studied in 1936 by Paul Erdős and
Pál Turán Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting ...
, who conjectured that they must be sparse. However, in 1942,
Raphaël Salem Raphaël Salem (Greek: Ραφαέλ Σαλέμ; November 7, 1898 in Salonika, Ottoman Empire (now Thessaloniki, Greece) – June 20, 1963 in Paris, France) was a Greek mathematician after whom are named the Salem numbers and Salem–Spencer sets ...
and Donald C. Spencer constructed progression-free subsets of the numbers from 1 to n of size proportional to n^, for every \varepsilon>0. Roth vindicated Erdős and Turán by proving that it is not possible for the size of such a set to be proportional to n: every
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
set of integers contains a three-term arithmetic progression. His proof uses techniques from analytic number theory including the
Hardy–Littlewood circle method In mathematics, the Hardy–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem. History The initial idea is usually at ...
to estimate the number of progressions in a given sequence and show that, when the sequence is dense enough, this number is nonzero. Other authors later strengthened Roth's bound on the size of progression-free sets. A strengthening in a different direction,
Szemerédi's theorem In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers ''A'' with positive natural density contains a ''k''-ter ...
, shows that dense sets of integers contain arbitrarily long arithmetic progressions.


Discrepancy

Although Roth's work on Diophantine approximation led to the highest recognition for him, it is his research on irregularities of distribution that (according to an obituary by William Chen and
Bob Vaughan Robert Charles "Bob" Vaughan FRS (born 24 March 1945) is a British mathematician, working in the field of analytic number theory. Life Since 1999 he has been Professor at Pennsylvania State University, and since 1990 Fellow of the Royal Soci ...
) he was most proud of. His 1954 paper on this topic laid the foundations for modern
discrepancy theory In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, name ...
. It concerns the placement of n points in a unit square so that, for every rectangle bounded between the origin and a point of the square, the area of the rectangle is well-approximated by the number of points in it. Roth measured this approximation by the squared difference between the number of points and n times the area, and proved that for a randomly chosen rectangle the expected value of the squared difference is logarithmic in n. This result is best possible, and significantly improved a previous bound on the same problem by
Tatyana Pavlovna Ehrenfest Tatyana Pavlovna Ehrenfest, later van Aardenne-Ehrenfest, (Vienna, October 28, 1905 – Dordrecht, November 29, 1984) was a Dutch mathematician. She was the daughter of Paul Ehrenfest (1880–1933) and Tatyana Alexeyevna Afanasyeva (1876–1964). ...
. Despite the prior work of Ehrenfest and
Johannes van der Corput Johannes Gaultherus van der Corput (4 September 1890 – 16 September 1975) was a Dutch mathematician, working in the field of analytic number theory. He was appointed professor at the University of Fribourg (Switzerland) in 1922, at the Univers ...
on the same problem, Roth was known for boasting that this result "started a subject".


Other topics

Some of Roth's earliest works included a 1949 paper on sums of powers, showing that almost all positive integers could be represented as a sum of a square, a cube, and a fourth power, and a
1951 Events January * January 4 – Korean War: Third Battle of Seoul – Chinese and North Korean forces capture Seoul for the second time (having lost the Second Battle of Seoul in September 1950). * January 9 – The Government of the United ...
paper on the gaps between
squarefree number In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...
s, describes as "quite sensational" and "of considerable importance" respectively by Chen and Vaughan. His inaugural lecture at Imperial College concerned the large sieve: bounding the size of sets of integers from which many
congruence class In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
es of numbers modulo
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 ways ...
s have been forbidden. Roth had previously published a paper on this problem in 1965. Another of Roth's interests was the
Heilbronn triangle problem In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed ...
, of placing points in a square to avoid triangles of small area. His
1951 Events January * January 4 – Korean War: Third Battle of Seoul – Chinese and North Korean forces capture Seoul for the second time (having lost the Second Battle of Seoul in September 1950). * January 9 – The Government of the United ...
paper on the problem was the first to prove a nontrivial upper bound on the area that can be achieved. He eventually published four papers on this problem, the latest in
1976 Events January * January 3 – The International Covenant on Economic, Social and Cultural Rights enters into force. * January 5 – The Pol Pot regime proclaims a new constitution for Democratic Kampuchea. * January 11 – The 1976 ...
. Roth also made significant progress on
square packing in a square Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side . If is an integer, the answer is , but the precise, or even asymptotic, amount ...
. If unit squares are packed into an s\times s square in the obvious, axis-parallel way, then for values of s that are just below an integer, nearly 2s area can be left uncovered. After Paul Erdős and
Ronald Graham Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He ...
proved that a more clever tilted packing could leave a significantly smaller area, only O(s^), Roth and
Bob Vaughan Robert Charles "Bob" Vaughan FRS (born 24 March 1945) is a British mathematician, working in the field of analytic number theory. Life Since 1999 he has been Professor at Pennsylvania State University, and since 1990 Fellow of the Royal Soci ...
responded with a 1978 paper proving the first nontrivial lower bound on the problem. As they showed, for some values of s, the uncovered area must be at least proportional In 1966,
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 Roth published their book ''
Sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
'', 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. For ...
s. Initially planned to be the first of a two-volume set, its topics included the densities of sums of sequences, bounds on the number of representations of integers as sums of members of sequences, density of sequences whose sums represent all integers,
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
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 ...
, and sequences in which no element is a multiple of another. A second edition was published in 1983.


Recognition

Roth won the Fields Medal in 1958 for his work on Diophantine approximation. He was the first British Fields medalist. He was elected to the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1960, and later became an Honorary Fellow of the Royal Society of Edinburgh, Fellow of University College London, Fellow of Imperial College London, and Honorary Fellow of Peterhouse. It was a source of amusement to him that his Fields Medal, election to the Royal Society, and professorial chair came to him in the reverse order of their prestige. The London Mathematical Society gave Roth the
De Morgan Medal The De Morgan Medal is a prize for outstanding contribution to mathematics, awarded by the London Mathematical Society. The Society's most prestigious award, it is given in memory of Augustus De Morgan, who was the first President of the society ...
in 1983. In 1991, the Royal Society gave him their
Sylvester Medal The Sylvester Medal is a bronze medal awarded by the Royal Society (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry a ...
"for his many contributions to number theory and in particular his solution of the famous problem concerning approximating algebraic numbers by rationals." A
festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the h ...
of 32 essays on topics related to Roth's research was published in 2009, in honour of Roth's 80th birthday, and in 2017 the editors of the journal ''
Mathematika ''Mathematika'' is a peer-reviewed mathematics journal that publishes both pure and applied mathematical articles. The journal was founded by Harold Davenport in the 1950s. The journal is published by the London Mathematical Society, on behalf of ...
'' dedicated a special issue to Roth. After Roth's death, the Imperial College Department of Mathematics instituted the Roth Scholarship in his honour.


Selected publications


Journal papers

* * * * * * * * *


Book

* A second edition was published in 1983 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 ...
.


Notes


References

{{DEFAULTSORT:Roth, Klaus 1925 births 2015 deaths 20th-century English mathematicians Academics of Imperial College London Alumni of Peterhouse, Cambridge Fellows of the Royal Society Fields Medalists De Morgan Medallists Number theorists People educated at St Paul's School, London People from the Province of Lower Silesia Jewish emigrants from Nazi Germany to the United Kingdom Alumni of University College London Academics of University College London