Szemeredi's Theorem states that any sufficiently dense subset
of the integers contains
arbitrarily long arithmetic progressions, and as such forms a crucial
ingredient in Green
and Tao's celebrated result on long arithmetic progressions in the primes.
At the heart
of the analytic proof of Szemeredi's Theorem lies the fact that if a subset
A of a finite
Abelian group G satisfies a quasi-randomness property called
"uniformity of degree k",
then it contains roughly the "expected" number of arithmetic progressions
of length k+2.
(By the "expected" number we mean the number of progressions one would
expect in a
random subset of G of the same density as A.)
One is naturally led to ask which degree of
uniformity is required of A in order to control the number of solutions
to a general system
of linear equations. Joint work with Tim Gowers has led to a precise
classification of such
systems according to a simple and easily verifiable criterion.