  Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term
arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and
N := p^n. A result of this type was claimed in a previous paper by the authors
and published in Proc. London Math. Society. Unfortunately the proof had a gap,
and we issue an erratum for that paper here. Our new argument is different and
significantly shorter. In fact we prove a stronger result, which can be viewed
as a quantatitive version of some previous results of Bergelson-Host-Kra and
the authors.
