Klaus Roth

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Klaus Roth
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Born
Klaus Friedrich Roth

(1925-10-29)29 October 1925
Died10 November 2015(2015-11-10) (aged 90)
Inverness, Scotland
Education
Known for
Awards
Scientific career
Fields Mathematics
Institutions
Thesis Proof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power (1950)
Doctoral advisor Theodor Estermann
Other academic advisors

Klaus Friedrich Roth FRS (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.

Fellow of the Royal Society Elected Fellow of the Royal Society, including Honorary, Foreign and Royal Fellows

Fellowship of the Royal Society is an award granted to individuals that the Royal Society of London judges to have made a 'substantial contribution to the improvement of natural knowledge, including mathematics, engineering science, and medical science'.

Fields Medal prize for mathematicians

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.

Roths theorem theorem that algebraic numbers do not have unusually accurate rational approximations

In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number may not have too many rational number approximations, that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).

Contents

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.

University of Cambridge University in Cambridge, United Kingdom

The University of Cambridge is a collegiate public research university in Cambridge, United Kingdom. Founded in 1209 and granted a Royal Charter by King Henry III in 1231, Cambridge is the second-oldest university in the English-speaking world and the world's fourth-oldest surviving university. The university grew out of an association of scholars who left the University of Oxford after a dispute with the townspeople. The two 'ancient universities' share many common features and are often referred to jointly as 'Oxbridge'. The history and influence of the University of Cambridge has made it one of the most prestigious universities in the world.

University College London, which has operated under the official name of UCL since 2005, is a public research university located in London, United Kingdom. It is a constituent college of the federal University of London, and is the third largest university in the United Kingdom by total enrolment, and the largest by postgraduate enrolment.

Imperial College London Public research university in London, United Kingdom

Imperial College London is a public research university located in London, England. In 1851, Prince Albert built his vision for a cultural area composed of the Victoria and Albert Museum, Natural History Museum, Royal Albert Hall, Royal Colleges, and the Imperial Institute. In 1907, Imperial College was established by Royal Charter, bringing together the Royal College of Science, Royal School of Mines, and City and Guilds College. In 1988, the Imperial College School of Medicine was formed through a merger with St Mary's Hospital Medical School. In 2004, Queen Elizabeth II opened the Imperial College Business School.

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. With Heini Halberstam he was the author of a book on integer sequences.

Salem–Spencer set

In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers. Salem-Spencer sets are named after Raphaël Salem and Donald C. Spencer, who showed in 1942 that Salem–Spencer sets can have nearly-linear size. However a later theorem of Klaus Roth shows that the size is always less than linear.

In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.

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, namely distributing points in some space such that they are evenly distributed with respect to some subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one.

As well as winning the Fields Medal, Roth was a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.

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.

Sylvester Medal bronze medal awarded by the Royal Society (London)

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 at the University of Oxford in the 1880s, and first awarded in 1901, having been suggested by a group of Sylvester's friends after his death in 1897. Initially awarded every three years with a prize of around £900, the Royal Society have announced that starting in 2009 it will be awarded every two years instead, and is to be aimed at 'early to mid career stage scientist' rather than an established mathematician. The award winner is chosen by the Society's A-side awards committee, which handles physical rather than biological science awards.

Royal Society English learned society for science

The President, Council and Fellows of the Royal Society of London for Improving Natural Knowledge, commonly known as the Royal Society, is a learned society. Founded on 28 November 1660, it was granted a royal charter by King Charles II as "The Royal Society". It is the oldest national scientific institution in the world. The society is the United Kingdom's and Commonwealth of Nations' Academy of Sciences and fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, fostering international and global co-operation, education and public engagement.

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. [1] [2] 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. [2]

Wrocław City in Lower Silesian Voivodeship, Poland

Wrocław is a city in western Poland and the largest city in the historical region of Silesia. It lies on the banks of the River Oder in the Silesian Lowlands of Central Europe, roughly 350 kilometres (220 mi) from the Baltic Sea to the north and 40 kilometres (25 mi) from the Sudeten Mountains to the south. The population of Wrocław in 2018 was 640,648, making it the fourth-largest city in Poland and the main city of the Wrocław agglomeration.

Prussia state in Central Europe between 1525–1947

Prussia was a historically prominent German state that originated in 1525 with a duchy centred on the region of Prussia on the southeast coast of the Baltic Sea. It was de facto dissolved by an emergency decree transferring powers of the Prussian government to German Chancellor Franz von Papen in 1932 and de jure by an Allied decree in 1947. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organised and effective army. Prussia, with its capital in Königsberg and from 1701 in Berlin, decisively shaped the history of Germany.

World War I 1914–1918 global war originating in Europe

World War I, also known as the First World War or the Great War, was a global war originating in Europe that lasted from 28 July 1914 to 11 November 1918. Contemporaneously described as "the war to end all wars", it led to the mobilisation of more than 70 million military personnel, including 60 million Europeans, making it one of the largest wars in history. It is also one of the deadliest conflicts in history, with an estimated nine million combatants and seven million civilian deaths as a direct result of the war, while resulting genocides and the 1918 influenza pandemic caused another 50 to 100 million deaths worldwide.

Mathematical education

Roth read mathematics at Peterhouse, Cambridge, and played first board for the Cambridge chess team, [2] finishing in 1945. [3] 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, was not supportive of Roth continuing in mathematics, recommending instead that he take "some commercial job with a statistical bias". [2] Nevertheless, he became a schoolteacher at Gordonstoun. [1] [2]

Peterhouse, Cambridge La universidad de los petes

Peterhouse is a constituent college of the University of Cambridge. It is the oldest college of the university, having been founded in 1284 by Hugo de Balsham, Bishop of Ely, and granted its charter by King Edward I. Today, Peterhouse has 254 undergraduates, 116 full-time graduate students and 54 fellows. The official name of Peterhouse does not include "college", although "Peterhouse College" is often seen in public.

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.

John Charles Burkill FRS was an English mathematician who worked on analysis and introduced the Burkill integral. He was elected a fellow of the Royal Society in 1953. In 1948, Burkill won the Adams Prize. He was Master of Peterhouse until 1973.

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. [2] He completed a master's degree there in 1948, and a doctorate in 1950. [3] His dissertation was Proof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power. [4]

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. [5] He was promoted to professor in 1961. [1] During this period, he continued to work closely with Harold Davenport. [2]

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 and Patrick Linstead countered this threat to British mathematics with an offer of a chair in pure mathematics at Imperial College London, which he accepted in 1966. [2] He retained this position until official retirement in 1988. [1] He remained at Imperial College as Visiting Professor until 1996. [3]

Roth's lectures were usually very clear but could occasionally be erratic. [2] He had few doctoral students, [4] but one of them, William Chen, became a Fellow of the Australian Mathematical Society and head of the mathematics department at Macquarie University. [6]

Personal life

In 1955, Roth married Mélèk Khaïry, a daughter of Egyptian senator Khaïry Pacha, who had attracted his attention as a student in his first lecture. [1] [2] Khaïry came to work for the psychology department at University College London, where she published research on the effects of toxins on rats. [7] On Roth's retirement, they moved to Inverness; Roth dedicated a room of their house to Latin dancing, a shared interest of theirs. [2] [8] Khaïry died in 2002, and Roth died in Inverness on 10 November 2015 at the age of 90. [1] [2] [3] 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. [9]

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". [10] 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 numbers 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 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 continued fractions. [3] [10] 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. [2]

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. [3] This result has been called Roth's "greatest achievement". [10]

Arithmetic combinatorics

The set {1,2,4,5,10,11,13,14} (blue) has no 3-term arithmetic progression, as the average of every two set members (yellow) falls outside the set. Roth proved that every progression-free set must be sparse. Salem-Spencer-8-14.svg
The set {1,2,4,5,10,11,13,14} (blue) has no 3-term arithmetic progression, as the average of every two set members (yellow) falls outside the set. Roth proved that every progression-free set must be sparse.

Another result called "Roth's theorem", from 1953, is a milestone in arithmetic combinatorics. It 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. [11] [lower-alpha 1] However, in 1942, Raphaël Salem and Donald C. Spencer constructed progression-free subsets of the numbers from to of size proportional to , for every . [12]

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 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. [2] [13]

Other authors later strengthened Roth's bound on the size of progression-free sets. [14] A strengthening in a different direction, Szemerédi's theorem, shows that dense sets of integers contain arbitrarily long arithmetic progressions. [15]

Discrepancy

The Hammersley set, a low-discrepancy set of points obtained from the van der Corput sequence Hammersley set 2D.svg
The Hammersley set, a low-discrepancy set of points obtained from the van der Corput sequence

Although Roth's work on Diophantine approximation led to the highest recognition for him, it is his research on irregularities of distribution "that gave him the greatest satisfaction". [2] His 1954 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. [2]

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 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. [16] 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". [2]

Other topics

Some of Roth's earliest works included a "quite sensational" 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 paper "of considerable importance" on the gaps between squarefree numbers. [2] 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 numbers have been forbidden. [17] Roth had previously published a paper on this problem in 1965.

The optimal square packing in a square can sometimes involve tilted squares; Roth and Bob Vaughan showed that non-constant area must be left uncovered 5 kvadratoj en kvadrato.svg
The optimal square packing in a square can sometimes involve tilted squares; Roth and Bob Vaughan showed that non-constant area must be left uncovered

Another of Roth's interests was the Heilbronn triangle problem, of placing points in a square to avoid triangles of small area. His 1951 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. [18] 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 , [19] Roth and Bob Vaughan responded with a 1978 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 to . [2] [20]

In 1966, 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. [21] A second edition was published in 1983. [22]

Recognition

The Fields Medal FieldsMedalFront.jpg
The Fields Medal

Roth won the Fields Medal in 1958 for his work on Diophantine approximation. He was the first British Fields medalist. [1] He was elected to the Royal Society 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. [1] 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. [2]

The London Mathematical Society gave Roth the De Morgan Medal in 1983. [3] 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." [23]

A festschrift of 32 essays on topics related to Roth's research was published in 2009, in honour of Roth's 80th birthday, [24] and in 2017 the editors of the journal Mathematika dedicated a special issue to Roth. [25] After Roth's death, the Imperial College Department of Mathematics instituted the Roth Scholarship in his honour. [26]

Selected publications

Journal papers

Book

Notes

  1. Davenport (1960) gives the date of the Erdős–Turán conjecture as 1935, but states that it "is believed to be older". He states the conjecture in the form that the natural density of a progression-free sequence should be zero, which Roth proved. However, the form of the conjecture actually published by Erdős & Turán (1936) is much stronger, stating that the number of elements from to in such a sequence should be for some exponent . In this form, the conjecture was falsified by Salem & Spencer (1942).

Related Research Articles

A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.

Powerful number integer where each prime number dividing it can be squared to still divide it

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

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-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.

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Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.

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References

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  2. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Chen, William; Vaughan, Robert (14 June 2017). "Klaus Friedrich Roth. 29 October 1925 — 10 November 2015". Biographical Memoirs of Fellows of the Royal Society. 63: 487–525. doi:10.1098/rsbm.2017.0014. ISSN   0080-4606. See also Chen, William; Larman, David; Stuart, Trevor; Vaughan, Robert (January 2016). "Klaus Friedrich Roth, 29 October 1925 – 10 November 2015". Newsletter of the London Mathematical Society via Royal Society of Edinburgh.
  3. 1 2 3 4 5 6 7 Jing, Jessie; Servini, Pietro (24 March 2015). "A Fields Medal at UCL: Klaus Roth". Chalkdust.
  4. 1 2 Klaus Roth at the Mathematics Genealogy Project
  5. O'Connor, John J.; Robertson, Edmund F., "Klaus Roth", MacTutor History of Mathematics archive , University of St Andrews .
  6. Chen, William Wai Lim. "Curriculum vitae" . Retrieved 25 April 2019.
  7. Khairy, Melek (May 1959). "Changes in behaviour associated with a nervous system poison (DDT)". Quarterly Journal of Experimental Psychology. 11 (2): 84–91. doi:10.1080/17470215908416295.Khairy, M. (April 1960). "Effects of chronic dieldrin ingestion on the muscular efficiency of rats". Occupational and Environmental Medicine. 17 (2): 146–148. doi:10.1136/oem.17.2.146.
  8. Szemerédi, Anna Kepes (2015). "Conversation with Klaus Roth". Art in the Life of Mathematicians. Providence, Rhode Island: American Mathematical Society. pp. 248–253. doi:10.1090/mbk/091. ISBN   978-1-4704-1956-1. MR   3362651.
  9. MacDonald, Stuart (26 April 2016). "Mathematician leaves £1m to help sick patients in Inverness". The Scotsman .
  10. 1 2 3 Davenport, H. (1960). "The work of K. F. Roth". Proc. Internat. Congress Math. 1958 (PDF). Cambridge University Press. pp. lvii–lx. MR   1622896. Zbl   0119.24901. Reprinted in Fields Medallists' Lectures (1997), World Scientific, pp. 53–56.
  11. Erdős, Paul; Turán, Paul (1936). "On some sequences of integers" (PDF). Journal of the London Mathematical Society . 11 (4): 261–264. doi:10.1112/jlms/s1-11.4.261. MR   1574918.
  12. Salem, R.; Spencer, D. C. (December 1942). "On sets of integers which contain no three terms in arithmetical progression". Proceedings of the National Academy of Sciences . 28 (12): 561–563. doi:10.1073/pnas.28.12.561.
  13. Heath-Brown, D. R. (1987). "Integer sets containing no arithmetic progressions". Journal of the London Mathematical Society. Second Series. 35 (3): 385–394. doi:10.1112/jlms/s2-35.3.385. MR   0889362.
  14. Bloom, T. F. (2016). "A quantitative improvement for Roth's theorem on arithmetic progressions". Journal of the London Mathematical Society . Second Series. 93 (3): 643–663. arXiv: 1405.5800 . doi:10.1112/jlms/jdw010. MR   3509957.
  15. Szemerédi, Endre (1975). "On sets of integers containing no k elements in arithmetic progression" (PDF). Acta Arithmetica . 27: 199–245. doi:10.4064/aa-27-1-199-245. MR   0369312. Zbl   0303.10056.
  16. van Aardenne-Ehrenfest, T. (1949). "On the impossibility of a just distribution". Indagationes Math. 1: 264–269. MR   0032717.
  17. Vaughan, Robert C. (December 2017). Diamond, Harold G. (ed.). "Heini Halberstam: some personal remarks". Heini Halberstam, 1926–2014. Bulletin of the London Mathematical Society. Wiley. 49 (6): 1127–1131. doi:10.1112/blms.12115. See page 1127: "I had attended Roth’s inaugural lecture on the large sieve at Imperial College in January 1968, and as a result had started to take an interest in sieve theory."
  18. Barequet, Gill (2001). "A lower bound for Heilbronn's triangle problem in d dimensions". SIAM Journal on Discrete Mathematics. 14 (2): 230–236. doi:10.1137/S0895480100365859. MR   1856009.. See the introduction, which cites the 1951 paper as "the first nontrivial upper bound" and refers to all four of Roth's papers on the Heilbronn triangle problem, calling the final one "a comprehensive survey of the history of this problem".
  19. Erdős, P.; Graham, R. L. (1975). "On packing squares with equal squares" (PDF). Journal of Combinatorial Theory . Series A. 19: 119–123. doi:10.1016/0097-3165(75)90099-0. MR   0370368.
  20. Brass, Peter; Moser, William; Pach, János (2005). Research Problems in Discrete Geometry. New York: Springer. p. 45. ISBN   978-0387-23815-9. MR   2163782.
  21. 1 2 Reviews of Sequences:
  22. 1 2 MR 0687978
  23. "Winners of the Sylvester Medal of the Royal Society of London". MacTutor History of Mathematics Archive. Retrieved 25 April 2019.
  24. Chen, W. W. L.; Gowers, W. T.; Halberstam, H.; Schmidt, W. M.; Vaughan, R. C., eds. (2009). "Klaus Roth at 80". Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press. ISBN   978-0-521-51538-2. Zbl   1155.11004.
  25. Chen, William W. L.; Vaughan, Robert C. (2017). "In memoriam Klaus Friedrich Roth 1925–2015". Mathematika. 63 (3): 711–712. doi:10.1112/S002557931700033X. MR   3731299.
  26. "PhD Funding opportunities". Imperial College London Department of Mathematics. Retrieved 26 April 2019.