Title: Fourier Analysis and Szemerédi's Theorem

The famous theorem of Szemer\'edi 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 $\{1,2,\dots,N\}$ 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 Szemer\'edi'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 Szemer\'edi's theorem which greatly improves the best known bound when $k=4$, and which will probably do the same for general $k$.

1991 Mathematics Subject Classification: 11P99

Keywords and Phrases: Arithmetic progressions, Fourier analysis

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