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

In this paper we prove a basic theorem which says that if f : Fpn → [0, 1] has few ‘large’ Fourier coefficients, and if the sum of the norms squared of the ‘small’ Fourier coefficients is ‘small’ (i.e. the L2 mass of f̂ is not concetrated on small Fourier coefficients), then Σn,df(n)f(n+ d)f(n + 2d) is ‘large’. 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−γ) for some small γ > 0. We conjecture that the condition that the sum of norms squared of small Fourier coefficients should not be necessary to obtain this conclusion; and, if this can be proved, it would mean that any subset of Fpn deficient in three-term progressions, must have quite of lot of large Fourier coefficients, even when the set has very low density.