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

By Ernie Croot


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 ˆ f 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.

Year: 2007
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