2006/07/07 by Ernie Croot, Croot, Ernie · 2 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #math.CO #math.NT #msc:11P70
paper · pdf · doi:10.48550/arxiv.math/0607209
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
In this paper we prove a basic theorem which says that if f : Fpn -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-δin place of 1/3), and E(f) = p-n summ 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 pn(1-c) for some small c > 0.