Ernie Croot
Published 2006-07-07, updated 2006-09-15Version 4
In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not too ``small'', then there are lots of triples m,m+d,m+2d such that f(m)f(m+d)f(m+2d) > 0. If f is the indicator function for some set S, then this would be saying that the set has many three-term arithmetic progressions. In principle this theorem can be applied to sets having very low density, where |S| is around p^{n(1-c)} for some small c > 0.
Comments: One small notational correction: In the paper I called ||f||_(1/3) a `norm', when in fact it should be 'quasinorm'. This does not affect any results, as I don't use the triangle inequality anywhere -- the 1/3 quasinorm was only used as a convenient way to state a corollary of one of my results
On the Structure of Sets with Few Three-Term Arithmetic Progressions
On systems of complexity one in the primes
On the maximal number of three-term arithmetic progressions in subsets of Z/pZ
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