N T ] 1 5 Se p 20 06 On the Decay of the Fourier Transform and Three Term Arithmetic Progressions

@inproceedings{Croot2007NT,
title={N T ] 1 5 Se p 20 06 On the Decay of the Fourier Transform and Three Term Arithmetic Progressions},
author={Ernie Croot},
year={2007}
}

In this paper we prove a basic theorem which says that if the tail of the spectral L norm of a function f : Fpn → [0, 1] is sufficiently small (i.e. the function f is “sufficiently smooth”), then there are lots of arithmetic progressions m,m+ d,m+ 2d where f(m)f(m+ d)f(m+ 2d) > 0. If f were an indicator function for some set S, then this would be saying that S has many three-term arithmetic progressions. In principle this theorem can be applied to sets having very low density, where |S| is… CONTINUE READING