Fourier analysis and Szemerédi's theorem

Abstract

The famous theorem of Szemerédi asserts that for every positive integer $k$ and every positive real number $\delta>0$ there is a positive integer $N$ such that every subset of $\{1,2,\cdots,N\}$ of cardinality at least $\delta N$ contains an arithmetic progression of length $k$. A second proof of the theorem was given by Furstenberg using ergodic theory, but neither this proof nor Szemerédi's gave anything other than extremely weak information about the dependence of $N$ on $k$ and $\delta$. In this article we describe a new, more quantitative approach to Szemerédi's theorem which greatly improves the best known bound when $k=4$, and which will probably do the same for general $k$. See also Geom. Funct. Anal. 8, 529-551 (1998; Zbl 0907.11005), A new proof of Szemerédi's theorem, ibid. 11, 465-588 (2001; Zbl 1028.11005).