Klaus Roth

Klaus Roth
Born	Klaus Friedrich Roth 29 10, 1925
Birthplace	Breslau, Province of Lower Silesia, Weimar Germany
Died	Template:Death date and age Inverness, Scotland, United Kingdom
Nationality	British
Occupation	Mathematician
Known for	Roth's theorem on Diophantine approximation, progression-free sets, irregularities of distribution
Education	PhD, University College London (1950)
Awards	Fields Medal (1958), De Morgan Medal (1983), Sylvester Medal (1991)

Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician whose work in number theory and combinatorics placed him among the most distinguished mathematicians of the twentieth century. He was the first British-based mathematician to receive the Fields Medal, awarded in 1958 for his proof that irrational algebraic numbers cannot be well-approximated by rational numbers — a result now universally known as Roth's theorem. Born in Breslau during the Weimar Republic, Roth fled Nazi Germany with his family in 1933 and settled in England, where he would spend the remainder of his life and career. After studying at the University of Cambridge and completing his doctorate at University College London, he held academic positions at University College London and Imperial College London, contributing to a broad range of mathematical fields including arithmetic combinatorics, the theory of irregularities of distribution, the large sieve method, and problems in discrete geometry. He was elected a Fellow of the Royal Society, a Fellow of the Royal Society of Edinburgh, and received both the De Morgan Medal and the Sylvester Medal. Upon his death in 2015, he left a substantial charitable bequest to support hospital patients in the Scottish Highlands.

Early Life

Klaus Friedrich Roth was born on 29 October 1925 in Breslau, then part of the Province of Lower Silesia in the Weimar Republic (now Wrocław, Poland). His family was of Jewish heritage, and as the political situation in Germany deteriorated following the rise of the Nazi Party, the Roth family made the decision to leave the country. In 1933, when Klaus was eight years old, the family emigrated to England to escape Nazi persecution.

Roth grew up in England and received his early education there. Despite the upheaval of emigration during childhood and the disruptions of the Second World War, he demonstrated exceptional mathematical ability from a young age. He attended school in London before proceeding to university studies. The experience of displacement and the broader context of wartime Britain shaped his formative years, though Roth himself was known to be reticent about his personal history, preferring to let his mathematical work speak for itself.

The circumstances of Roth's emigration were shared by a number of other mathematicians and scientists who fled Central Europe during the 1930s and went on to make significant contributions in Britain and elsewhere. Roth's trajectory from refugee child to Fields Medalist represents one of the notable stories of intellectual migration in twentieth-century mathematics.

Education

Roth studied mathematics at the University of Cambridge, where he received his undergraduate education. He subsequently moved to University College London (UCL) for his postgraduate work. At UCL, Roth pursued research under the supervision of Theodor Estermann, a noted analytic number theorist. He completed his PhD in 1950.

His time at Cambridge and UCL exposed him to the British tradition of analytic number theory, which had been shaped by figures such as G. H. Hardy, J. E. Littlewood, and Harold Davenport. These influences would prove formative for Roth's subsequent research, which combined classical techniques of analytic number theory with innovative new approaches. His doctoral work at UCL laid the groundwork for the remarkable series of results he would produce in the 1950s, including the theorem that would earn him the Fields Medal.

Career

University College London (1948–1966)

Roth began his academic career at University College London, where he was appointed to a teaching position even before completing his doctorate. He joined the mathematics department at UCL in 1948 and would remain there for nearly two decades. It was during this period that Roth produced the work for which he is best known.

In 1955, Roth proved his celebrated theorem on Diophantine approximation, which demonstrated that for any irrational algebraic number α and any ε > 0, there are only finitely many rational numbers p/q satisfying the inequality |α − p/q| < q^−2−ε. In other words, irrational algebraic numbers cannot be approximated by rationals to a degree better than exponent 2 (plus an arbitrarily small constant). This result settled a longstanding conjecture that had been the subject of successive improvements by Axel Thue, Carl Ludwig Siegel, and Freeman Dyson, and it represented the definitive solution to a problem with roots stretching back to the work of Joseph Liouville in the nineteenth century.

The significance of Roth's theorem cannot be understated in the history of number theory. Previous results by Thue (1909), Siegel (1921), and Dyson (1947) had progressively lowered the exponent from which algebraic irrationals could be approximated, but none had achieved the optimal exponent of 2. Roth's proof achieved this optimal bound, completing a line of research that had occupied some of the finest mathematical minds for over a century. The theorem had profound consequences for the study of Diophantine equations, as it implied that many such equations could have only finitely many solutions.

Also during his time at UCL, Roth made a major contribution to arithmetic combinatorics. In 1953, he proved that any subset of the natural numbers with positive upper density must contain a three-term arithmetic progression. This result, sometimes also referred to as Roth's theorem (on arithmetic progressions), was a landmark in combinatorial number theory and represented the first non-trivial case of the conjecture that would later be proved in full generality by Endre Szemerédi in 1975 as Szemerédi's theorem. Roth's proof employed an innovative application of the Hardy–Littlewood circle method to a combinatorial problem, a technique that opened new avenues of research in the field.

Roth's method for proving the existence of three-term arithmetic progressions built upon earlier work by Paul Erdős and Pál Turán, who had conjectured this result in 1936. Where previous approaches had failed to make progress on the conjecture, Roth's analytic method provided a decisive breakthrough that established the result once and for all for progressions of length three.

The Fields Medal (1958)

In 1958, at the International Congress of Mathematicians held in Edinburgh, Roth was awarded the Fields Medal for his work on Diophantine approximation. He was 32 years old at the time and became the first mathematician working in Britain to receive the prize, which is often described as the highest honour in mathematics.

The Fields Medal citation highlighted Roth's theorem on the approximation of algebraic numbers by rationals as the primary achievement for which the award was given. The award brought international recognition to the British number theory community and to University College London in particular. Roth's receipt of the medal was a source of considerable pride for UCL, which had not previously been associated with such a high-profile mathematical honour.

Roth presented an address at the 1958 Congress outlining his work and its implications for the theory of Diophantine equations. The proceedings of that Congress document the mathematical community's recognition of the depth and significance of his contribution.

Imperial College London (1966–1988)

In 1966, Roth left University College London to take up a chair in mathematics at Imperial College London, where he would spend the remainder of his active academic career.

During his time at Imperial, Roth continued to make important contributions across several areas of mathematics. One of his major areas of research was the theory of irregularities of distribution, sometimes called discrepancy theory. This field concerns the question of how evenly a finite set of points can be distributed in a given region. Roth proved fundamental results showing that perfect uniformity is impossible — that any finite point set must exhibit a certain minimum level of irregularity. His work in this area established basic lower bounds that remain central to the theory and influenced subsequent developments by many other mathematicians.

Roth also made contributions to the study of sums of powers, investigating the representation of integers as sums of powers of natural numbers. This work connected to the classical Waring's problem and its generalizations. His research on the large sieve — a powerful tool in analytic number theory used to obtain information about the distribution of prime numbers and other arithmetic objects — was influential in the development of modern sieve methods.

Another area in which Roth made notable contributions was discrete geometry. He worked on the Heilbronn triangle problem, which asks for the minimum area of the smallest triangle formed by n points placed in a unit square, and on the problem of square packing in a square — the question of how efficiently small squares of various sizes can be packed into a larger square. These problems, while seemingly elementary in their statements, involve deep mathematical ideas and have connections to combinatorics and optimization.

Together with Heini Halberstam, Roth co-authored the book Sequences, a monograph on integer sequences that became an important reference work in the field. The book provided a systematic treatment of various problems concerning the properties and distribution of sequences of integers, and it was recognized as a significant contribution to the mathematical literature.

Roth supervised a number of doctoral students during his career at Imperial College, contributing to the training of the next generation of number theorists and combinatorialists. Among his doctoral students was William Chen, who would go on to his own academic career in mathematics.

Roth retired from Imperial College London in 1988, having held his chair for over two decades.

Later Years

Following his retirement from Imperial College, Roth withdrew from active mathematical research. He and his wife eventually settled in Inverness, in the Scottish Highlands, where he lived quietly for the final years of his life. Despite his retirement from academic life, his earlier work continued to influence developments in number theory, combinatorics, and discrete geometry. New proofs and extensions of Roth's theorems appeared regularly in the mathematical literature throughout the 1990s and 2000s, attesting to the enduring importance of his contributions.

Roth was elected a Fellow of the Royal Society of Edinburgh during his later years in Scotland, reflecting his continued standing in the mathematical community even after retirement.

Klaus Roth died on 10 November 2015 in Inverness, at the age of 90.

Personal Life

Roth was known for his modesty and reluctance to seek the public spotlight, characteristics noted by colleagues and in obituary notices. Despite holding mathematics' most prestigious prize, he maintained a quiet and unassuming demeanor throughout his career.

After his retirement, Roth and his wife moved to Inverness in the Scottish Highlands. He lived there for many years, largely out of the public eye. Following his death in November 2015, it was reported that Roth had left approximately £1 million in his will to support sick patients at Raigmore Hospital in Inverness — a generous bequest that reflected his connection to his adopted community in the Highlands. This charitable act brought renewed public attention to Roth's name, this time in a context far removed from abstract mathematics.

Roth's personal papers and the details of his private life remained largely undisclosed. He was known to be a private individual who preferred to be remembered for his mathematical work rather than for personal or biographical details.

Recognition

Roth's mathematical achievements were recognized with several of the highest honours available to mathematicians.

His most prominent award was the Fields Medal, received in 1958 at the International Congress of Mathematicians in Edinburgh. This made him the first mathematician based in Britain to receive the prize, and he remained one of a very small number of British Fields Medalists for decades thereafter.

In 1960, Roth was elected a Fellow of the Royal Society (FRS), one of the highest scientific honours in the United Kingdom. Fellowship of the Royal Society recognized the significance and originality of his contributions to mathematics.

In 1983, the London Mathematical Society awarded Roth the De Morgan Medal, the Society's most prestigious prize, given every three years in recognition of outstanding mathematical achievement. His election to membership of the London Mathematical Society was also noted as a mark of the esteem in which he was held by the British mathematical community.

In 1991, the Royal Society awarded Roth the Sylvester Medal, given for outstanding contributions to mathematical research. The Sylvester Medal is one of the oldest and most distinguished prizes in British mathematics, and Roth's name was added to a list of recipients that includes some of the most eminent mathematicians in history.

Roth was also elected a Fellow of the Royal Society of Edinburgh, further recognizing his stature in the scientific community of his adopted country.

Legacy

Klaus Roth's contributions to mathematics have had a lasting and profound impact on several branches of the discipline. His theorem on Diophantine approximation, which established the optimal exponent for the approximation of irrational algebraic numbers by rationals, remains one of the landmark results of twentieth-century number theory. The theorem completed a programme of research that had engaged mathematicians for over a century and continues to be a foundational result in the field. Extensions and generalizations of Roth's theorem — including the Subspace theorem proved by Wolfgang M. Schmidt — have become central tools in modern Diophantine geometry.

Roth's work on arithmetic progressions was equally influential. His proof that sets of positive upper density contain three-term arithmetic progressions was the first significant step toward Szemerédi's theorem and, more broadly, toward the modern field of additive combinatorics. The techniques he introduced — particularly the application of Fourier-analytic methods (the circle method) to combinatorial problems — have been developed and extended by subsequent generations of mathematicians, including Timothy Gowers, Ben Green, and Terence Tao. The study of bounds in Roth's theorem on arithmetic progressions remains an active area of research, with improvements to the quantitative bounds continuing to appear in the mathematical literature.

His contributions to discrepancy theory established foundational results on the irregularities of distribution that continue to guide research in the area. The lower bounds he proved on the discrepancy of point distributions are among the classical results of the theory and have connections to computational complexity, numerical integration, and quasi-random sequences.

The co-authored book Sequences with Halberstam served as an influential reference in the study of integer sequences and their properties, contributing to the systematization of knowledge in this area of number theory.

At Imperial College London, the mathematics department has continued to honour Roth's legacy, and his name is associated with the tradition of excellence in number theory at the institution. Imperial College's PhD programme in mathematics, which Roth helped to build during his tenure, continues to attract research students from around the world.

Roth's charitable bequest to Raigmore Hospital in Inverness, totalling approximately £1 million, ensured that his legacy extended beyond the mathematical world and into the community that had been his home in retirement.