Klaus Friedrich Roth, first British winner of the Fields Medal, elected to membership of the LMS and awarded the De Morgan Medal, has died aged 90.

Klaus Friedrich Roth, who was elected to membership of the London Mathematical Society on 17 May 1951 and awarded the De Morgan Medal in 1983, has died in Inverness, aged 90. He was the first British winner of the Fields Medal, and made fundamental contributions to different areas of number theory, including Diophantine approximation, the large sieve, irregularities of distribution and arithmetic combinatorics.

*W.W.L. Chen, D.G. Larman, J.T. Stuart and R.C. Vaughan write:*

Klaus Roth was born on 29 October 1925, in the German city of Breslau, in Lower Silesia, Prussia, now Wroclaw in Poland. To escape from Nazism, he and his parents moved to England in 1933, with his maternal grandparents, and settled in London. He would recall that the flight from Berlin to London took eight hours and landed in Croydon. His father had suffered from gas poisoning during the First World War, and died within a few years of their arrival in England.

Roth studied at St Paul's School, and proceeded to read mathematics at the University of Cambridge, where he was a student at Peterhouse and also played first board for the university chess team. However, he had many unhappy and painful memories of his two years there as an undergraduate. Uncontrollable nerves was to seriously hamper his examination results, and he graduated with third class honours.

After this not too distinguished start to his academic career, Roth then did his war time service as an alien and became a junior master at Gordonstoun, where he divided his spare time between roaming the Scottish countryside on a powerful motorcycle and playing chess with Robert Combe. On the first day of the first British Chess Championships after the war, Klaus famously went up to Hugh Alexander, the reigning champion, to tell him that he would not retain his title. He was of course right -- the previously largely unknown Robert Combe became the new British Champion.

Peterhouse did not support Roth’s return to Cambridge after war service, and his tutor J.C. Burkill had suggested instead that he pursued “some commercial job with a statistical bias”. However, his real ability and potential, particularly his problem solving skills, had not escaped the eyes of Harold Davenport, who subsequently arranged for him to pursue mathematical research at University College London, funded by the highest leaving exhibition ever awarded by his old school. Although Theodor Estermann was officially his thesis advisor, Roth was heavily influenced by Davenport during this period, and indeed into the mid 1960s. He completed his PhD work which Estermann considered good enough for a DSc, and also joined the staff of the Department of Mathematics.

The early problems Roth studied were centred around the Hardy-Littlewood technique, the highlight being undoubtedly his novel application of the technique to sequences that are not explicitly given and resulting in his famous result on three terms in arithmetic progression, paving the way for further progress by Szemerédi, Gowers, Green, Tao and others in arithmetic combinatorics.

Davenport’s influence clearly cultivated Roth’s interest in Diophantine approximation. There is significant work already done by Dirichlet, Liouville, Thue, Siegel, Dyson and Gelfond. A crucial exponent was believed to depend on the degree of the algebraic number under consideration, but Siegel had conjectured that it should be 2. This is precisely what Roth showed. In a letter to Davenport, Siegel commented that this result “will be remembered as long as mankind is interested in mathematics”. For this and his result on three terms in arithmetic progression, Roth was awarded the Fields Medal in 1958.

In speaking of Roth's work at the Opening Ceremony of the International Congress of Mathematicians in 1958, Davenport said, “The achievement is one that speaks for itself: it closes a chapter, and a new chapter is opened. Roth's theorem settles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated,” and ended with the following words. “The Duchess, in Alice in Wonderland, said that there is a moral in everything if only you can find it. It is not difficult to find the moral of Dr Roth's work. It is that the great unsolved problems may still yield to direct attack however difficult and forbidding they appear to be, and however much effort has already been spent on them.”

While most mathematicians consider Roth’s result on Diophantine approximation as his most famous, it is in fact another problem that gives him the greatest satisfaction. At about the same time, he became interested in the question of the impossibility of a just distribution for any sequence in the unit interval, conjectured by van der Corput in 1935. Van Aardenne-Ehrenfest obtained the first quantitative estimate in 1949. By reformulating the problem in a geometric setting, Roth obtained in 1954 the best possible lower bound for the mean squares of the discrepancy function. This geometric setting paved the way for what is now known as geometric discrepancy theory, a subject at the crossroads of harmonic analysis, combinatorics, approximation theory, probability theory and even group theory. Once asked why he considered this his best work, Roth replied, “But I started a subject!” He was particularly pleased that J.C. Burkill, who had remained on good terms, offered the same opinion.

Further recognition came. Roth was elected Fellow of the Royal Society in 1960 and also promoted to a professorship at the University of London in 1961. He was very proud that the Fields Medal, the Fellowship of the Royal Society and the professorship came somewhat in reverse order.

The close relationship between Roth and Davenport in those days can be illustrated by a charming incident some time in the 1950s and which Heini Halberstam recalled with great delight. Early one Sunday morning, Davenport went to his bathroom and switched on the light. The phone rang, and it was Roth. Could he possibly come over and explain the proof of a new result? Davenport suggested that Roth came after breakfast, but as soon as he put the phone down, the door bell rang. Roth had been so eager that he had spent much of the early morning waiting in the telephone booth across the street.

It was also at this time at University College London that Roth met his wife Melek Khaïry. It was the first ever university lecture given by him and the first ever university lecture attended by her. After the lecture, Klaus had asked Halberstam whether he had noticed the young lady on the front row. “I will marry her,” he claimed. By the end of that year, Roth had felt unable to mark Melek’s examination script, claiming that he felt “unable to be impartial”, much to the amusement of his colleagues. Another problem at the time was that during their courtship, Melek's sister Hoda often came along. To counter that, Roth brought along his best friend at the time as a distraction for Hoda. His friend took to his assignment with great gusto, and indeed married Hoda.

In the mid 1960s, Roth produced his seminal paper on the large sieve. This is a major step towards better understanding of primes in arithmetic progressions, and the beginning of a major development in analytic number theory.

At about the same time, he had planned to emigrate to the United States to take up the offer of a position at the Massachusetts Institute of Technology where he had spent a year a decade earlier. Imperial College intervened, and agreement was reached in the middle of a reception at the Soviet Embassy in London. Roth said that Sir Patrick Linstead, then Rector of Imperial College, told him that he needed to make an application, but that there would be no other applicant. So Roth joined Imperial College in 1966 after a sabbatical at the MIT, and remained there until his retirement.

At Imperial College, Roth continued an extensive study of the irregularities of integer sequences relative to arithmetic progressions and also made substantial progress on Heilbronn’s triangle problem. His famous ¼-theorem of 1964 is the basis of a Fourier transform approach to irregularities of distribution by Beck in the 1980s, leading to spectacular results. His last work concerns the introduction of probability theory into the study of upper bounds in irregularities of distribution, paving the way for many applications in numerical integration, with implications in many branches of science, engineering and finance.

Roth moved with Melek to Inverness after his retirement. Melek’s death in 2002 was a great setback, and Roth never recovered from this loss.

Roth was an excellent lecturer. He explained things so clearly that a good student could often just sit there and listen, and only had to record the details afterwards in the evening. However, he occasionally would have an off day, and he warned his students at the beginning of the year that they would notice these very easily. One of us recalls that on one occasion, Roth wrote down a very complicated expression on the blackboard, then retired to the back of the room. A lot of thought was followed by an equal sign, and he retired to the back of the room again. After a long time he came once more up to the board and wrote down the same complicated expression on the right hand side. The audience held their collective breath at this profound assertion. But the best was yet to come. He then proceeded to write down + O(1), at which point all burst out laughing. Roth looked at his masterpiece again, turned to the class and protested, “But it is correct, isn't it?”

Outside mathematics, Roth enjoyed ballroom dancing, and would waltz away the evening elegantly with Melek. They took this very seriously, to the point that they had a room in their Inverness house specially fitted for dancing practice. For many years while they were in London, they had dancing lessons with Alan Fletcher, who with his wife Hazel were then world ballroom dancing champion. Indeed, Roth dedicated one of his research papers to Alan. He explained that he had been bothered by a problem which he could not solve and was therefore not dancing very well, and that Alan had annoyed him so much by asking him week after week without fail whether he had solved his problem. So when he finally managed to crack it, he needed to acknowledge Alan for having provided the annoyance.

Roth had maintained great modesty throughout his life. He felt very privileged to have been given the opportunity to pursue what he loved, and very lucky that he had some moderate success. He had always been very generous to his colleagues, and had inspired many to achieve good results.

Roth also received the Sylvester Medal of the Royal Society in 1991. He was also elected Honorary Fellow of the Royal Society of Edinburgh in 1993, Fellow of University College London in 1979 and of Imperial College London in 1999, and an Honorary Fellow of Peterhouse in 1989.

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**Claudia Cannon**

The Grantham Institute for Climate Change

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