Klaus Friedrich Roth, 1925-2015

Abstract

Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a member of the London Mathematical Society on 17 May 1951, and received its De Morgan Medal in 1983. Klaus Roth, son of Franz and Matilde (née Liebrecht), was born on 29 October 1925, in the German city of Breslau, in Lower Silesia, Prussia, now Wrocław in Poland. To escape from Nazism, he and his parents moved to England in 1933 and settled in London. He would recall that the flight from Berlin to London took eight hours and landed in Croydon. Franz, a solicitor by training, had suffered from gas poisoning during the First World War, and died a few years after their arrival in England. Roth studied at St Paul's School between 1937 and 1943, during which time the school was relocated to Easthampstead Park, near Crowthorne in Berkshire, as part of the wartime evacuation of London. There he excelled in mathematics and chess, and one master, Mr Dowswell, observed interestingly that he possessed complete intellectual honesty. As extracurricular activity, Roth was deeply interested in the Air Training Corps, but his efforts to be a member were thwarted for a long time because of his German nationality, until special permission was finally given towards the end of his time at St Paul's. His badly coordinated muscular movements ensured that his wish to become a pilot was never going to be realized. Roth proceeded to read mathematics at the University of Cambridge, and became a student at Peterhouse. He also played first board for the university chess team. However, he had many unhappy and painful memories of his two years in Cambridge as an undergraduate. Uncontrollable nerves would 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, Roth famously went up to Hugh Alexander, the reigning champion, to tell him that he would not retain his title. This 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 his war service, and his tutor John Charles Burkill had suggested instead that he pursued ‘some commercial job with a statistical bias’. Fortunately, 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 then 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. Davenport's influence clearly cultivated Roth's interest in diophantine approximation. Significant work had already been done by Dirichlet, Liouville, Thue, Siegel, Dyson and Gelfond. Indeed, 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. In 1955, Roth showed precisely that. In a letter to Davenport, Siegel commented that this result ‘will be remembered as long as mankind is interested in mathematics’. For this, 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 gave 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 Burkill, with whom he 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 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 should come 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 during this time at University College London that Roth met his wife Melek Khaïry. Melek and her sister Hoda, daughters of the senator Khaïry Pacha in Egypt, had defied the wishes of their old-fashioned family apart from their father and come to study in London. It was the first ever university lecture given by Roth and the first ever university lecture attended by Melek. After the lecture, Roth 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 unsuitable 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, Hoda often tagged along, much to Roth's annoyance. To counter that, Roth brought along his best friend Laimons Ozolins, the Latvian born architect and fellow pupil at St Paul's School, as a distraction for Hoda. Ozolins took to his assignment with great gusto, and indeed married Hoda. In the mid-1960s, Roth 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 and Walter Hayman intervened, and agreement was reached in the middle of a reception at the Soviet Embassy in London. Roth recalled that Sir Patrick Linstead, then Rector of Imperial College, told him that he needed to make an application, but reassured him 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. Following his retirement in 1987, Roth moved with Melek to Inverness. Melek's death in 2002 was a great setback, and Roth never recovered from this loss. In later years, Roth became increasingly disappointed at the services and facilities available to old people in Inverness, and subsequently left the bulk of his estate towards improving these. Roth was an excellent lecturer. He explained his points 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 a bad 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 into laughter. Roth looked at his masterpiece again, turned to the class and protested, ‘But it is correct, isn't it?' Outside mathematics, Roth enjoyed Latin American dancing, and would elegantly jive away the evening with Melek. They took this very seriously, to the point that they had a room in their house in Inverness specially fitted for dancing practice. For many years while they were in London, they had dancing lessons with Alan Fletcher, who, with wife Hazel, was five-time world Latin American dancing champion. Indeed, Roth dedicated one of his research papers to Fletcher. He explained that he had been bothered by a problem which he could not solve and was therefore not dancing very well, and that Fletcher had annoyed him so much by asking him week after week without fail whether he had solved his problem. So to get Fletcher off his back, he just had to crack the problem, and when he did, he needed to acknowledge Fletcher for having provided the annoyance. Roth 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 De Morgan Medal of the London Mathematical Society in 1983 and 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. He and Melek had no children. Roth's work in the very early part of his career concerns the application of the Hardy–Littlewood method to study certain additive questions in number theory. His subsequent work can be described as a career long fascination with and repeated efforts at understanding the limitations to the degree of regularity possible in various discrete systems, often making very clever use of artificial orthogonality or quasi-orthogonality, and punctuated by a small number of spectacular digressions, including his seminal contribution to diophantine approximation 14 and to the large sieve 21. In the mid-1950s, with encouragement and advice from Paul Erdős, Roth and his close colleague Heini Halberstam began to write the influential volume Sequences 23. The effort took nearly ten years, and Roth particularly enjoyed writing about Rényi's version of the large sieve. He worked on it even after the completion of the book, culminating in his own remarkable contribution to the subject. Halberstam recalled fondly that Roth carried the completed manuscript by hand all the way to the offices of the Clarendon Press in Oxford. Questions on regularity occupy the bulk of Roth's writings, and these can be divided roughly into three areas: irregularities of integer sequences in arithmetic progressions, irregularities of point distribution and Heilbronn's triangle problem. Notation.Throughout this article, we adopt the O, o as well as Vinogradov notation ≪ and ≫. Thus for any function f and any positive function g, we write f = O ( g ) or f ≪ g to denote that there exists a positive constant c such that | f | ⩽ c g , and write f = o ( g ) to denote that | f / g | → 0 . Furthermore, if f is also a positive function, then we write f ≫ g to denote g ≪ f , and write f ≍ g to indicate that f ≪ g and f ≫ g both hold. The symbols O, o, ≪ and ≫ may have subscripts if the constant c in question depends on the variables represented by those subscripts. The cardinality of a finite set A is denoted by | A | . A squarefree number is a positive integer with no repeated prime factors. They have some properties analogous to prime numbers but are generally less demanding to understand, and thus often provide a good testing ground for techniques. Estermann 〈〈39〉〉 had obtained an asymptotic formula for the number of representations of a large natural number as the sum of a square and a squarefree number. In his first research paper Roth 1 extended this result to the situation in which the square was restricted to being the square of a squarefree number. Whilst not a technically demanding problem, it nevertheless provided a good introduction to the methodology applied to questions in analytic number theory. His second paper 4 on squarefree numbers is of considerable importance, and deals with gaps between squarefree numbers. There followed papers by Richert 〈〈98〉〉, Rankin 〈〈86〉〉, Schmidt 〈〈104〉〉, Graham and Kolesnik 〈〈50〉〉, Trifonov 〈〈123〉, 〈124〉〉 and Filaseta 〈〈41〉〉, many of them quite technical. On the other hand, in collaboration with Halberstam, Roth developed his original ideas to treat k-free numbers, namely, those integers with no more than k − 1 repeated prime factors, in their paper 5. There is also a substantial literature on gaps between k-free numbers, and also on k-free values of polynomials, much of it stimulated by the ideas in 5. For an article with a good overview of the subject, see Filaseta 〈〈42〉〉. The Hardy–Littlewood method is a technique in additive number theory, developed in the 1920s from a famous paper of Hardy and Ramanujan 〈〈66〉〉 by Hardy and Littlewood in a series of papers 〈〈58〉, 〈59〉, 〈60〉, 〈62〉, 〈61〉, 〈63〉, 〈64〉, 〈65〉〉. There had been many important developments by Davenport and I. M. Vinogradov, and Estermann was also considered a leading expert on the method. Thus it is not surprising that, with both Davenport and Estermann as mentors at University College London, several chapters in Roth's PhD thesis should involve applications of the method. Roth had been given by Davenport the problem of showing that almost every natural number n could be expressed as the sum of a square, a cube, a fourth power and a fifth power, in the sense that the number of exceptional n ⩽ N is o ( N ) . He met Estermann one day and announced that actually he did not need the fifth power. This result in 2 was quite sensational. At the time there had been relatively little done on ternary additive problems. Of course, there was the classical theorem of Gauss–Legendre that every natural number not of the form 4 j ( 8 k + 7 ) could be expressed as the sum of three squares. There were also a couple of papers by Davenport and Heilbronn. In 〈〈32〉〉, they showed that almost every natural number could be expressed as the sum of two squares and a kth power with k odd. In 〈〈31〉〉, they showed that almost every natural number could be expressed as the sum of a square and two cubes. However, these were the extent of the known results. Thus Roth's theorem pushed the envelope of what was known. A further necessary condition for solubility of these equations is that they be soluble modulo q for every modulus q, and this is essentially the same as requiring that there be a non-singular solution in each p-adic field. In particular, since there are a positive proportion of n which cannot be represented as the sum of three squares, we may suppose in the case 3.2 that k 3 is odd. We now know that in many of these configurations, there are infinitely many natural numbers n for which local solubility is not sufficient. The first such examples are due to Jagy and Kaplansky 〈〈69〉〉; see also Vaughan 〈〈129〉, Chapter 8, Exercise 5〉. More recently, others have been added to the list by Dietmann and Elsholtz 〈〈35〉, 〈36〉〉 and by Gundlach 〈〈54〉〉. Apparently, in each case, the counter-example can be interpreted as a Brauer–Manin obstruction. However, these failures of the local to global principle only occur for a thin set of natural numbers n, and this enhances the interest of results which show solubility for almost all natural numbers n, and in particular in estimates for the size of any exceptional set. It has been possible to show in each case 3.2–3.5 that almost all natural numbers n can be represented in the form 3.1. As mentioned above the first results of this kind are due to Davenport and Heilbronn, and Roth added to this with his work on a square, a cube and a fourth power. The picture was completed by Vaughan 〈〈127〉〉 who established that almost all natural numbers could be expressed as the sum of a square, a cube and a fifth power. There is a considerable body of work on adapting these methods to situations in which one or more of the variables are restricted in some way. See Vaughan 〈〈132〉〉 for a review of this material. Davenport had used the Hardy–Littlewood method to show in 〈〈27〉〉 that every sufficiently large natural number could be expressed as the sum of eight cubes and, more significantly, that almost every natural number could be expressed as the sum of four cubes. Roth then showed in 6 that in each case all but one of the variables could be taken to be prime. This is somewhat routine, although there are some technical difficulties to be overcome. However this paper clearly led to an interest in Vinogradov's methods for estimating exponential sums and resulted in the translation into English by Roth and Anne Davenport of Vinogradov's monograph 〈〈134〉〉 on exponential sums. Roth added extensive notes to each of the chapters, and Vinogradov told him at the International Congress of Mathematicians in Edinburgh in 1958 that serious consideration should be given to translating the book back into Russian! The work on cubes also attracted a large body of modern work, leading, for example, to Kawada's result in 〈〈70〉〉 that the non-prime variable could be replaced by a number having at most three prime factors. It would be of great interest if the non-prime could be replaced by a prime. There is a brief survey of this area in Vaughan 〈〈131〉〉. When η = 0 and k = 2 , Davenport and Heilbronn 〈〈33〉〉 had shown that s = 5 was permissible and it was clear that their method would establish the desired conclusion for general k and η with s = 2 k + 1 . In their paper, Davenport and Roth obtained a result with s ( k ) satisfying lim sup s ( k ) / ( k log k ) = 6 . For joint work by two of the most powerful analytic number theorists of the era, this is a surprisingly ordinary result. In dealing with the somewhat more onerous situation in which the variables were assumed to be prime, Vaughan 〈〈126〉〉 was able to obtain s ( k ) with lim sup s ( k ) / ( k log k ) = 4 . Later, applying the techniques developed in Waring's problem by Vaughan 〈〈128〉〉 and Wooley 〈〈136〉〉, Li 〈〈74〉〉 was able to obtain the desired conclusion with an s ( k ) satisfying lim sup s ( k ) / ( k log k ) = 1 . The method used here depends on rational approximations a / q to one of the irrational ratios, for example λ r / λ s , as given by the continued fraction expansion. In particular, the range for the variables x 1 , … , x s depends on the size of q. However, suitably good rational approximations a / q can be very rare, so that the denominator q n of the nth convergent can grow extremely rapidly as a function of n. This can happen if, for example, the ratio is a Liouville number. Thus the method does not allow one to localize the solutions, in the sense that given a large parameter X, one cannot guarantee that there is a solution with, say, X < max j x j ⩽ X . There has been a considerable blossoming of work in this area since the problem of localization of solutions was overcome in a ground breaking paper of Bentkus and Götze 〈〈12〉〉. This was followed by papers by Freeman 〈〈46〉〉 and Wooley 〈〈137〉〉. A comprehensive review of this area is given in the paper of Brüdern, Kawada and Wooley 〈〈17〉〉. There is one other paper which can be considered in the classical Hardy–Littlewood method milieu. The paper 13 with Szekeres on generalized partition functions has been largely overlooked. It restricts to the case when the summands in a partition are distinct but the method applies more generally. In particular the method is easily adapted to give an asymptotic formula for the number of partitions of a large number into primes, a result which until quite recently, with the appearance of Vaughan 〈〈130〉〉, experts in the area had thought could not be obtained in the current state of knowledge. It can be said that Roth's early work on the Hardy–Littlewood method is not his most important work. Yet it is unlikely that, without this introductory phase, Roth would have considered using a variant of the Hardy–Littlewood method to treat sets having no three terms in arithmetic progression, with all that which followed. See Section 7. Questions of diophantine approximation have attracted the attention of the world's leading mathematicians for at least four centuries, and continue to do so. Roth's theorem on diophantine approximation settled a central question which had already been heavily worked over by leading researchers. To appreciate his achievement, it is necessary to give a short review of the earlier work. It suffices to suppose that α is an algebraic integer of degree d ⩾ 2 . The first step is the index theorem. This states that if α is an algebraic integer, k ⩾ d / 2 ε 2 and e 1 , … , e k are positive integers, then there is a polynomial of the form 4.2, not identically zero, such that P has degree at most e j in the variable x j , I ( α , e ) ⩾ k ( 1 − ε ) / 2 and H ( P ) ⩽ C ( α ) e 1 + ⋯ + e k . The core of the argument is Roth's lemma. This shows that if P is as in 4.2 and satisfies H ( P ) ⩽ q 1 e 1 θ , where θ < 1 depends at most on k and ε, then I ( ρ , e ) ⩽ ε . The proof is based on the properties of generalized Wronskians. The proof is completed by using Roth's lemma to obtain a contradiction against the index theorem. The method was extended by Baker 〈〈4〉〉 to show, for example, that the transcendental (Champernowne) number 0.1234567890111213 … is not a U-number in the sense of Mahler. The large sieve was introduced by Linnik 〈〈75〉〉 with the aim of considering questions in which, at least on average, one could sieve out a large number of residue classes modulo each prime p. This work was then developed by Rényi in a long series of papers 〈〈87〉, 〈88〉, 〈89〉, 〈90〉, 〈91〉, 〈92〉, 〈93〉, 〈94〉, 〈95〉, 〈96〉〉; see also 〈〈97〉〉. Perhaps the most important application made by Rényi was as an aid to small sieves which led to theorems of the kind that every sufficiently large even number could be expressed as the sum of a prime and a number having a bounded number of prime factors. Rényi seemed to have been seduced by the probabilistic nature of the sum V ( q ) , but this did not really lead to any further profound advances. The state of the art before the seminal papers of Roth 21 and Bombieri 〈〈15〉〉 was summarized by Barban 〈〈5〉, 〈6〉〉. At this point, the methods were quite effective when Q ⩽ N 1 / 3 , but less effective for larger values of Q. However, by the time the survey article by Barban 〈〈6〉〉 appeared, it was obsolescent. Roth essayed only one paper in convexity and the geometry of numbers. This is especially surprising since this was a major interest of Davenport, and Roth was ideally equipped to work in the area. However, Rogers was already making great progress and undoubtedly Roth was only too happy to let him get on with it. However, a visit by Bambah stimulated his interest. There are two intriguing questions that involve trigonometrical polynomials with integer frequencies. Later, following progress by Pichorides and Fournier, Konyagin 〈〈73〉〉 and McGehee, Pigno and Smith 〈〈81〉〉 independently and by different methods proved Littlewood's conjecture 6.1. For a comprehensive survey of the area, see Odlyzko 〈〈84〉〉. With regard to the Ankeny–Chowla problem, it is still an open question as to how large one can take f ( N ) in 6.2. Chowla 〈〈21〉〉 conjectured that there might even be a positive constant c such that f ( N ) = c N 1 / 2 would hold. If true, this would be best possible; see Pichorides 〈〈85〉〉. The story behind this is that, some time in the late spring of 1977, Hugh Montgomery had mentioned to one of us [Vaughan] the question of packing unit squares into a large square of side length ℓ not equal to an integer, say ℓ = n + θ where 0 < θ < 1 . Perhaps not surprisingly, just lining up as many unit squares as possible with the sides of the large square, which leaves an area uncovered of 2 n θ + θ 2 , is not very efficient, at least when θ is not very small, and rather more efficient packings are known. For example, Chung and Graham 〈〈22〉〉 could exhibit a packing with the uncovered area at most C ℓ ϕ log ℓ , where ϕ = ( 3 + 2 ) / 7 = 0.6306 … . The more interesting question is whether one can show that a certain amount of space can never be covered however ingenious the packing. The Imperial College Mathematics Department held its annual examiners meeting in June, which went on all day. That year it was, as usual, quite tedious. One had to stay alert in case something came up about one's own tutees, and occasionally there would be some discussion about borderline or exceptional cases, but otherwise it could be a bit of a bore. Anyway, at lunch that day, in order to lighten the mood, the problem was described to Roth. He visibly brightened, and a few days later came up with a very clever argument which eventually led to a joint paper 35 showing that the amount of waste space, whatever the packing, was always at least c ( n min { θ , 1 − θ } ) 1 / 2 . Afterwards he claimed that this had been a much more interesting examiners meeting than usual! The principal idea is to think of a ray entering the large square from the left and to suppose that on reaching the side of a unit square it is ‘refracted’ to a direction orthogonal to that face. If the little squares through which it passes are not skewed very much, then it will have to pass through waste space to a distance of roughly θ. If there is an appreciable amount of skewing, then there will be triangular pieces of waste between successive squares the area of which can be approximated in terms of the angles of skew. Thus the amount of waste space can be bounded below by a sum of cosines. There are complications if rays cross each other, but in principle one can obtain a lower bound for the amount of waste area. The question then arises whether 1 / 2 is the right exponent. Probably it is not. The proof was ‘one dimensional’. If one could take advantage of both dimensions, then perhaps the exponent could be made larger. Nevertheless the theorem is still the best that is known. Roth wrote eight papers 8, 10, 11, 20, 24, 25, 27, 28 on the distribution of sequences of natural numbers in arithmetic progressions and related subjects. They were motivated by the following fundamental result established in 1927. Theorem. (van der Waerden 〈[〈135〉]〉)If the natural numbers are partitioned into any finite number of classes, then at least one of the classes must contain arbitrarily long arithmetic progressions. In particular, he was fascinated by the following conjecture made in 1936. Conjecture. (Erdős and Turán 〈[〈38〉]〉)If a strictly increasing sequence of natural numbers has positive upper density, then it contains arbitrarily long arithmetic progressions. The solution of this conjecture in 1975 is one of the cornerstones of combinatorics. Theorem. (Szemerédi 〈[〈120〉]〉)Let r k ( N ) denote the greatest number of natural numbers that can be selected from 1 , … , N to form a set that does not contain any arithmetic progression of length k. Then r k ( N ) = o k ( N ) as N → ∞ . It is worthwhile to note that Roth's 1 / 4 -theorem is sharp, as shown by Matoušek and Spencer 〈〈79〉〉. On the other hand, one-sided discrepancy problems, seeking to establish the existence of an arithmetic progression A for which D [ S ; A ] is large and non-negative, is of particular interest. Although this was not discussed in his paper in 1964, Roth 40 commented in a survey written in 2000 that the following was equivalent to the conjecture of Erdős and Turán, nowadays known as Szemerédi's theorem. Proposition.Let c ∈ ( 0 , 1 / 2 ) be given. Then there exists ε > 0 such that for all sufficiently larg