Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician who won the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
for proving
Roth's theorem 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 ...
of
algebraic numbers. He was also a winner of the
De Morgan Medal and the
Sylvester Medal, 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 university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...
and
University College London
University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...
, 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, also known as Imperial, is a Public university, public research university in London, England. Its history began with Prince Albert of Saxe-Coburg and Gotha, Prince Albert, husband of Queen Victoria, who envisioned a Al ...
. He retired in 1988.
Beyond his work on Diophantine approximation, Roth made major contributions to the theory of
progression-free sets in
arithmetic combinatorics 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, and on
square packing in a square. He was a coauthor of the book ''
Sequences'' 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: ''Prūsija'') was a Germans, German state centred on the North European Plain that originated from the 1525 secularization of the Prussia (region), Prussian part of the State of the Teutonic Order. For centuries, ...
, 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 or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...
and died while Roth was still young. Roth became a pupil at
St Paul's School, London from 1939 to 1943, and with the rest of the school he was evacuated from London to
Easthampstead Park during
the Blitz
The Blitz (English: "flash") was a Nazi Germany, German bombing campaign against the United Kingdom, for eight months, from 7 September 1940 to 11 May 1941, during the Second World War.
Towards the end of the Battle of Britain in 1940, a co ...
. At school, he was known for his ability in both chess and mathematics. He tried to join the
Air Training Corps, 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 on the
Mathematical Tripos
The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics, University of Cambridge, Faculty of Mathematics at the University of Cambridge.
Origin
In its classical nineteenth-century form, the tripos was a di ...
, 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, 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 and education
Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accringto ...
, he was accepted in 1946 to a master's program in mathematics at
University College London
University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...
, where he worked under the supervision of
Theodor Estermann. 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 university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
in the mid-1950s and mid-1960s, and seriously considered migrating to the United States.
Walter Hayman and
Patrick Linstead 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, also known as Imperial, is a Public university, public research university in London, England. Its history began with Prince Albert of Saxe-Coburg and Gotha, Prince Albert, husband of Queen Victoria, who envisioned a Al ...
, 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 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 and head of the mathematics department at
Macquarie University
Macquarie University ( ) is a Public university, public research university in Sydney, New South Wales, Australia. Founded in 1964 by the New South Wales Government, it was the third university to be established in the Sydney metropolitan area. ...
.
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 is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
discrepancy theory, 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 ...
seeks accurate approximations of
irrational number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
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 (for example,
The set of all ...
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-called w ...
and
Carl Ludwig Siegel. The accuracy of approximation can be measured by the
approximation exponent of a number
, defined as the largest number
such that
has infinitely many rational approximations
with
. If the approximation exponent is large, then
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
simple continued fractions. 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, 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 and education
Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accringto ...
to the
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
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", from
1953
Events
January
* January 6 – The Asian Socialist Conference opens in Rangoon, Burma.
* January 12 – Estonian émigrés found a Estonian government-in-exile, government-in-exile in Oslo.
* January 14
** Marshal Josip Broz Tito ...
, is in
arithmetic combinatorics and concerns
sequences of integers with no three in arithmetic progression. These sequences had been studied in 1936 by
Paul Erdős and
Pál Turán, who conjectured that they must be sparse.
However, in 1942,
Raphaël Salem and
Donald C. Spencer constructed progression-free subsets of the numbers from
to
of size proportional to
, for every
.
Roth vindicated Erdős and Turán by proving that it is not possible for the size of such a set to be proportional to
: every
dense set of integers contains a three-term arithmetic progression. His proof uses techniques from
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
including the
Hardy–Littlewood circle method 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, 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) he was most proud of. His
1954
Events
January
* January 3 – The Italian broadcaster RAI officially begins transmitting.
* January 7 – Georgetown–IBM experiment: The first public demonstration of a machine translation system is held in New York, at the head ...
paper on this topic laid the foundations for modern
discrepancy theory. It concerns the placement of
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
times the area, and proved that for a randomly chosen rectangle the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of the squared difference is logarithmic in
. This result is best possible, and significantly improved a previous bound on the same problem by
Tatyana Pavlovna Ehrenfest. Despite the prior work of Ehrenfest and
Johannes van der Corput 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
Events
January
* January 1 – A United Nations-sponsored ceasefire brings an end to the Indo-Pakistani War of 1947. The war results in a stalemate and the division of Kashmir, which still continues as of 2025
* January 2 – Luis ...
paper on
sums of powers, showing that
almost all
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
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 Uni ...
paper on the gaps between
squarefree numbers, 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 classes of numbers modulo
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s have been forbidden. Roth had previously published a paper on this problem in
1965
Events January–February
* January 14 – The First Minister of Northern Ireland and the Taoiseach of the Republic of Ireland meet for the first time in 43 years.
* January 20
** Lyndon B. Johnson is Second inauguration of Lynd ...
.

Another of Roth's interests was the
Heilbronn triangle problem, 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 Uni ...
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 2 – 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 18 – Full diplomatic ...
.
Roth also made significant progress on
square packing in a square. If unit squares are packed into an
square in the obvious, axis-parallel way, then for values of
that are just below an integer, nearly
area can be left uncovered. After
Paul Erdős and
Ronald Graham proved that a more clever tilted packing could leave a significantly smaller area, only
, Roth and
Bob Vaughan responded with a
1978
Events January
* January 1 – Air India Flight 855, a Boeing 747 passenger jet, crashes off the coast of Bombay, killing 213.
* January 5 – Bülent Ecevit, of Republican People's Party, CHP, forms the new government of Turkey (42nd ...
paper proving the first nontrivial lower bound on the problem. As they showed, for some values of
, the uncovered area must be at least proportional
In
1966
Events January
* January 1 – In a coup, Colonel Jean-Bédel Bokassa takes over as military ruler of the Central African Republic, ousting President David Dacko.
* January 3 – 1966 Upper Voltan coup d'état: President Maurice Yaméogo i ...
,
Heini Halberstam and Roth published their book ''
Sequences'', 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 and the
probabilistic method, and
sequences in which no element is a multiple of another. A second edition was published in 1983.
Recognition

Roth won the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
in 1958 for his work on Diophantine approximation. He was the first British Fields medallist. 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
The Royal Society of Edinburgh (RSE) is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was establis ...
, 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
The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
gave Roth the
De Morgan Medal in 1983.
In 1991, the Royal Society gave him their
Sylvester Medal "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
University College London
University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...
journal ''
Mathematika'' 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 in ...
.
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
British 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