Density-increment proof of the finite-field four-term progression theorem

If a dense set has too few four-term arithmetic progressions, its balanced function has large $U^3$ norm. A Frequency graph extracted from a large Gowers U3 norm, followed by the Balog-Szemerédi-Gowers theorem and a Freiman theorem, yields correlation with a quadratic phase. Restricting to a suitable level set gives a density increment, and iteration proves the Finite-field Szemerédi theorem for four-term arithmetic progressions.