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{\bf Terence Tao }
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{\bf A Quantitative Ergodic Theory Proof of Szemer\'edi's Theorem}
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A famous theorem of Szemer\'edi asserts that given any density $0 <
\delta \leq 1$ and any integer $k \geq 3$, any set of integers with
density $\delta$ will contain infinitely many proper arithmetic
progressions of length $k$. For general $k$ there are essentially
four known proofs of this fact; Szemer\'edi's original combinatorial
proof using the Szemer\'edi regularity lemma and van der Waerden's
theorem, Furstenberg's proof using ergodic theory, Gowers' proof using
Fourier analysis and the inverse theory of additive combinatorics, and
the more recent proofs of Gowers and R\"odl-Skokan using a hypergraph
regularity lemma. Of these four, the ergodic theory proof is arguably
the shortest, but also the least elementary, requiring passage (via
the Furstenberg correspondence principle) to an infinitary measure
preserving system, and then decomposing a general ergodic system
relative to a tower of compact extensions. Here we present a
quantitative, self-contained version of this ergodic theory proof, and
which is ``elementary'' in the sense that it does not require the
axiom of choice, the use of infinite sets or measures, or the use of
the Fourier transform or inverse theorems from additive combinatorics.
It also gives explicit (but extremely poor) quantitative bounds.
\bye