  It is well-known that if a subset A of a finite Abelian group G satisfies a
quasirandomness property called uniformity of degree k, then it contains
roughly the expected number of arithmetic progressions of length k, that is,
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. Using so-called "quadratic Fourier analysis", we show that
certain linear systems that were previously thought to require quadratic
uniformity are in fact governed by linear uniformity. More generally, we
conjecture a necessary and sufficient condition on a linear system L which
guarantees that any subset A of F_p^n which is uniform of degree k contains the
expected number of solutions to L.
