= 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>.
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