Klaus Friedrich Roth. 29 October 1925 — 10 November 2015

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 Fellow of the Royal Society in 1960, and received its Sylvester Medal in 1991. He was also awarded the De Morgan Medal of the London Mathematical Society in 1983, and elected Fellow of University College London in 1979, Honorary Fellow of Peterhouse in 1989, Honorary Fellow of the Royal Society of Edinburgh in 1993 and Fellow of Imperial College London in 1999.

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[16]  Freeman J. Dyson,et al.  The approximation to algebraic numbers by rationals , 1947 .

[17]  Jörg Brüdern.  A problem in additive number theory , 1988 .

[18]  K. F. Roth.  On a Problem of Heilbronn, III , 1972 .

[19]  A. Rényi.  On the Representation of an Even Number as the Sum of a Prime and an Almost Prime , 1984 .

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[24]  K. F. Roth.  On irregularities of distribution , 1954 .

[25]  N. Ankeny.  The least quadratic non residue , 1952 .

[26]  T. Wooley.  On diophantine inequalities: Freeman's asymptotic formulae , 2003 .

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[30]  E. Szemeri~di,et al.  On Sets of Integers Containing No Four Elements in Arithmetic Progression , .

[31]  K. F. Roth.  Irregularities of sequences relative to arithmetic progressions, IV , 1972 .

[32]  I. Chowla.  On waring’s problem for cubes , 1937 .

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[47]  Paul Erdös,et al.  On Some Sequences of Integers , 1936 .

[48]  K. F. Roth.  On cosine polynomials corresponding to sets of integers , 1973 .

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[51]  R. Dietmann,et al.  Sums of two squares and one biquadrate , 2008 .

[52]  Ben Green,et al.  New bounds for Szemeredi's theorem, II: A new bound for r_4(N) , 2006 .

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[57]  C. Stewart.  Irregularities of sequences relative to long arithmetic progressions by A . Sárközy and , 2007 .

[58]  K. F. Roth.  Irregularities of sequences relative to arithmetic progressions , 1967 .

[59]  J. Beck.  On a problem of K.F. Roth concerning irregularities of point distribution , 1983 .

[60]  Some remarks concerning irregularities of distribution of sequences of integers in arithmetic progressions, III , 1978 .

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[67]  SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES , 2005, math/0506067.

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[69]  R. Vaughan.  Squares: Additive questions and partitions , 2015 .

[70]  C. A. Rogers.  A Note on Coverings and Packings , 1950 .

[71]  K. F. Roth,et al.  The solubility of certain Diophantine inequalities , 1955 .

[72]  Kevin Ford.  The representation of numbers as sums of unlike powers. II , 1996 .

[73]  H. Richert.  On the Difference between Consecutive Squarefree Numbers , 1954 .

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[76]  G. Szekeres,et al.  SOME ASYMPTOTIC FORMULAE IN THE THEORY OF PARTITIONS (II) , 1953 .

[77]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[78]  R. Dietmann,et al.  Sums of Two Squares and a Power , 2016, 1607.02614.

[79]  A. Baker.  On Mahler's classification of transcendental numbers , 1964 .

[80]  J. Vaaler.  SOME EXTREMAL FUNCTIONS IN FOURIER ANALYSIS , 2007 .

[81]  Primes in tuples I , 2005, math/0508185.

[82]  S. K. Zaremba,et al.  The extreme and L2 discrepancies of some plane sets , 1969 .

[83]  J. Beck,et al.  Note on Irregularities of Distribution II , 1990 .

[84]  D. E. Freeman.  Asymptotic lower bounds for diophantine inequalities , 2000 .

[85]  J. Maynard.  Small gaps between primes , 2013, 1311.4600.

[86]  K. F. Roth.  Proof that almost all Positive Integers are Sums of a Square, a Positive Cube and a Fourth Power , 1949 .

[87]  G. Hardy,et al.  Asymptotic Formulaæ in Combinatory Analysis , 1918 .

[88]  Axel Thue.  Über Annäherungswerte algebraischer Zahlen. , 1909 .

[89]  Fan Chung Graham,et al.  Packing equal squares into a large square , 2009, J. Comb. Theory, Ser. A.

[90]  On the least quadratic non-residue , 2008 .

[91]  K. F. Roth.  On a Problem of Heilbronn , 1951 .

[92]  T. Wooley,et al.  ADDITIVE REPRESENTATION IN THIN SEQUENCES, VIII: DIOPHANTINE INEQUALITIES IN REVIEW , 2009 .

[93]  Harold Davenport,et al.  A note on Thue's theorem , 1968 .

[94]  J. G. Corput.  Zahlentheoretische Abschätzungen mit Anwendung auf Gitterpunktprobleme , 1923 .

[95]  Enrico Bombieri,et al.  On the Large Sieve Method , 1969 .

[96]  K. F. Roth,et al.  The Method of Trigonometrical Sums in the Theory of Numbers , 2004 .

[97]  S. Chowla.  Some Applications of a Method of A. Selberg. , 1965 .

[98]  O. Carruth McGehee,et al.  Hardy's inequality and the L1 norm of exponential sums , 1981 .

[99]  M. Lacey,et al.  On the Small Ball Inequality in All Dimensions , 2007, 0705.4619.

[100]  K. F. Roth.  On the large sieves of Linnik and Rényi , 1965 .

[101]  H. Montgomery.  Ten lectures on the interface between analytic number theory and harmonic analysis , 1994 .

[102]  Wolfgang M. Schmidt.  Irregularities of distribution. IV , 1969 .

[103]  J. Beck,et al.  Irregularities of distribution , 1987 .

[104]  Robert C. Vaughan,et al.  A new iterative method in Waring's problem , 1989 .

[105]  ON GAPS BETWEEN SQUAREFREE NUMBERS , 1992 .

[106]  K. F. Roth,et al.  Rational approximations to algebraic numbers , 1955 .

[107]  E. Szemerédi,et al.  A Lower Bound for Heilbronn'S Problem , 1982 .

[108]  G. Hardy,et al.  Asymptotic formulae in combinatory analysis , 1918 .

[109]  A. Odlyzko.  Minima of Cosine Sums and Maxima of Polynomials on the Unit Circle , 1982 .

[110]  K. F. Roth.  Remark concerning integer sequences , 1964 .

[111]  C. A. Rogers.  Lattice Coverings of Space: The Minkowski–Hlawka Theorem , 1958 .

[112]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[113]  K. F Roth.  Irregularities of sequences relative to arithmetic progressions, III , 1970 .

[114]  C. Zong,et al.  A note on lattice coverings , 2014, 1402.4965.

[115]  D. R. Heath-Brown.  Integer Sets Containing No Arithmetic Progressions , 1987 .

[116]  J. Beck,et al.  IRREGULARITIES OF POINT DISTRIBUTION RELATIVE TO HALF-PLANES I , 1993 .

[117]  G. H. Hardy,et al.  Some problems of 'Partitio numerorum'; I: A new solution of Waring`s problem , 1920 .

[118]  W. M. Schmidt,et al.  Masstheorie in der Geometrie der Zahlen , 1959 .

[119]  Wolfgang M. Schmidt,et al.  Irregularities of distribution. VIII , 1974 .

[120]  K. F. Roth,et al.  Inefficiency in Packing Squares with Unit Squares , 1978, J. Comb. Theory, Ser. A.

[121]  Harold Davenport,et al.  Note on a Result in the Additive Theory of Numbers , 1938 .

[122]  Trevor D. Wooley,et al.  Large improvements in Waring's problem , 1992 .

[123]  Robert C. Vaughan,et al.  A Ternary Additive Problem , 1980 .

[124]  K. F. Roth,et al.  On the Gaps between Consecutive k‐Free Integers , 1951 .

[125]  G. H. Hardy,et al.  Some problems of ‘Partitio numerorum’ (VI): Further researches in Waring's Problem , 1925 .

[126]  H. Davenport.  Note on irregularities of distribution , 1956 .

[127]  C. Siegel.  Approximation algebraischer Zahlen , 1921 .

[128]  P. J. Cohen.  On a Conjecture of Littlewood and Idempotent Measures , 1960 .