three-term arithmetic progression. His proof uses techniques from analytic number theory including the Hardy–Littlewood circle method to estimate the number of progressions in a given sequence and show that, when the sequence is dense enough, this number is nonzero. Other authors later strengthened Roth's bound on the size of progression-free sets. A strengthening in a different direction, Szemerédi's theorem, shows that dense sets of integers contain arbitrarily long arithmetic progressions. Discrepancy Although Roth's work on Diophantine approximation led to the highest recognition for him, it is his research on irregularities of distribution that (according to an obituary by William Chen and Bob Vaughan) he was most proud of. His 1954 paper on this topic laid the foundations for modern discrepancy theory. It concerns the placement of points in a unit square so that, for every rectangle bounded between the origin and a point of the square, the area of the rectangle is well-approximated by the number of points in it. Roth measured this approximation by the squared difference between the number of points and times the area, and proved that for a randomly chosen rectangle the expected value of the squared difference is logarithmic in . This result is best possible, and significantly improved a previous bound on the same problem by Tatyana Pavlovna Ehrenfest. Despite the prior work of Ehrenfest and Johannes van der Corput on the same problem, Roth was known for boasting that this result "started a subject". Other topics Some of Roth's earliest works included a 1949 paper on sums of powers, showing that almost all positive integers could be represented as a sum of a square, a cube, and a fourth power, and a 1951 paper on the gaps between squarefree numbers, describes as "quite sensational" and "of considerable importance" respectively by Chen and Vaughan. His inaugural lecture at Imperial College concerned the large sieve: bounding the size of sets of integers from which many congruence classes of numbers modulo prime numbers have been forbidden. Roth had previously published a paper on this problem in 1965. Another of Roth's interests was the Heilbronn triangle problem, of placing points in a square to avoid triangles of small area. His 1951 paper on the problem was the first to prove a nontrivial upper bound on the area that can be achieved. He eventually published four papers on this problem, the latest in 1976. Roth also made significant