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

In this paper we prove a basic theorem which says that if the tail of the spectral $L^2$ norm of a function $f colon F_p^n o [0, 1]$ 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 around $p^{n(1 gamma)}$ for some small $gamma > 0.$ Furthermore, we show that if $g : F_p^n o [0,1]$ is majorized by $f$, and $mathbb{E}(g)$ is not too “small”, then in fact there are lots of progressions $m, m + d, m + 2d$ where $f(m)g(m + d)f(m + 2d) > 0.$