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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 the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
for proving Roth's theorem on the Diophantine approximation of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society. Roth moved to England as a child in 1933 to escape the Nazis, and was educated at the University of Cambridge and University College London, finishing his doctorate in 1950. He taught at University College London until 1966, when he took a chair at Imperial College London. 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 In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's thre ...
, on 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 only ...
, on the Heilbronn triangle problem, and on square packing in a square. He was a coauthor of the book '' Sequences'' on integer sequences.


Biography


Early life

Roth was born to a Jewish family in Breslau, Prussia, 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 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. 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 Peterhouse is the oldest constituent college of the University of Cambridge in England, founded in 1284 by Hugh de Balsham, Bishop of Ely. Today, Peterhouse has 254 undergraduates, 116 full-time graduate students and 54 fellows. It is quite ...
, 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, because of his poor test-taking ability. His Cambridge tutor,
John Charles Burkill John Charles Burkill (1 February 1900, Holt, Norfolk, England – 6 April 1993, Sheffield, England) was an English mathematician who worked on analysis and introduced the Burkill integral. He was educated at St Paul's School and Trinity College ...
, 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, he was accepted in 1946 to a master's program in mathematics at University College London, 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 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 ...
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, 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 research university based in Sydney, Australia, in the suburb of Macquarie Park. Founded in 1964 by the New South Wales Government, it was the third university to be established in the metropolitan area of S ...
.


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 Inverness (; from the gd, Inbhir Nis , meaning "Mouth of the River Ness"; sco, Innerness) is a city in the Scottish Highlands. It is the administrative centre for The Highland Council and is regarded as the capital of the Highlands. Histori ...
; 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, discrepancy theory, and the theory of integer sequences.


Diophantine approximation

The subject of Diophantine approximation seeks accurate approximations of irrational numbers by rational numbers. The question of how accurately
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s could be approximated became known as the Thue–Siegel problem, after previous progress on this question by Axel Thue and Carl Ludwig Siegel. The accuracy of approximation can be measured by the
approximation exponent In number theory, a Liouville number is a real number ''x'' with the property that, for every positive integer ''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that :0 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists ...
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 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 Events January * January 3 – José Ramón Guizado becomes president of Panama. * January 17 – , the first nuclear-powered submarine, puts to sea for the first time, from Groton, Connecticut. * January 18– 20 – Battle of Yijian ...
, 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 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 Nevanlinna Prize (to be rename ...
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 i ...
, 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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
and Pál Turán, who conjectured that they must be sparse. However, in 1942, Raphaël Salem and
Donald C. Spencer Donald Clayton Spencer (April 25, 1912 – December 23, 2001) was an American mathematician, known for work on deformation theory of structures arising in differential geometry, and on several complex variables from the point of view of pa ...
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 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 Diric ...
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 1 – The Soviet Union ceases to demand war reparations from West Germany. * January 3 – The Italian broadcaster RAI officially begins transmitting. * January 7 – Georgetown-IBM experiment: The fir ...
paper on this topic laid the foundations for modern discrepancy theory. 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 In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
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. 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 2022. * January 2 – Luis ...
paper on
sums of powers In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's thre ...
, showing that
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
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 numbers, describes as "quite sensational" and "of considerable importance" respectively by Chen and Vaughan. His inaugural lecture at Imperial College concerned 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 only ...
: bounding the size of sets of integers from which many congruence classes of numbers modulo prime numbers have been forbidden. Roth had previously published a paper on this problem in
1965 Events January–February * January 14 – The Prime 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 Lyndo ...
. 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 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 Phila ...
. Roth also made significant progress on square packing in a square. 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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
and Ronald Graham proved that a more clever tilted packing could leave a significantly smaller area, only O(s^), 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 go ...
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 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 sequences. 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 the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1958 for his work on Diophantine approximation. He was the first British Fields medalist. He was elected to the Royal Society in 1960, and later became an Honorary Fellow of the
Royal Society of Edinburgh The Royal Society of Edinburgh 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 established i ...
, 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 societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
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 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'' 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.


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