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MR1631259 (2000d:11019) 11B25; 11N13 Gowers, W. T. A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geometric and Functional Analysis 8 (1998), no. 3, 529–551. This remarkable paper gives a new proof that every subset of the integers with positive density must contain arithmetic progressions of length four. This was conjectured by P. Erdős and P. Turán [J. London Math. Soc. 11 (1936), 261– 264; Zbl 015.15203], and eventually proved by E. Szemerédi [Acta Math. Acad. Sci. Hungar. 20 (1969), 89–104; MR0245555]. Szemerédi’s proof depended on the theorem of van der Waerden. It gave no explicit function η(N), tending to zero with N , such that an integer subset of [1, N ] of size at least η(N)N must contain a progression of length 4. Since then it has been an important open problem, which this paper solves, to prove Szemerédi’s result with an explicit function η(N). The result given here is that there is a positive constant c such that any integer subset of [1, N ] with at least N(log log logN)−c elements contains an arithmetic progression of length 4. In a future paper it is promised that the number of logarithms will be reduced to 2. Moreover, the author plans to extend the technique to arithmetic progressions of arbitrary length k ≥ 4, obtaining bounds of the same form, but with the constant c depending on k. The initial stages of the proof are motivated by K. F. Roth’s treatment [J. London Math. Soc. 28 (1953), 104–109; MR0051853] of progressions of length 3, in which the circle method was used. However, it is clear that the usual notion of uniform distribution is not sufficient to handle progressions of length 4. Instead Gowers uses “quadratic uniformity”. If A ⊆ [1, N ] is an integer set with ηN elements, and characteristic function χA, define an = χA(n)−η. Roughly speaking, one says that A is quadratically uniform if ∑