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 , 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 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. A second edition was published in 1983. Recognition Roth won the Fields Medal in 1958 for his work on Diophantine approximation. He was the first British Fields medalist. 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. 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. The London Mathematical Society gave Roth the De Morgan Medal in 1983. 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." A festschrift of 32 essays on topics related to Roth's research was published in 2009, in honour of Roth's 80th birthday, and in 2017 the editors of the journal Mathematika dedicated a special issue to Roth. After Roth's death, the Imperial College Department of Mathematics instituted the Roth Scholarship in his honour. Selected publications Journal papers Book A second edition was published in 1983 by Springer-Verlag. Notes References 1925 births 2015 deaths 20th-century English mathematicians Academics of Imperial College London Alumni of Peterhouse, Cambridge Fellows of the Royal Society Fields Medalists De Morgan Medallists Number theorists People educated at St