On the Decay of the Fourier Transform and Three Term Arithmetic Progressions

Details On the Decay of the Fourier Transform and Three Term Arithmetic Progressions On the Decay of the Fourier Transform and Three Term Arithmetic Progressions

2006-07-07

arxiv.org/abs/math/0607209v4

Mathematics

Number Theory

One small notational correction: In the paper I called ||f||_(1/3) a `norm', when in fact it should be 'quasinorm'. This does

Scientific paper

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.

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