= Solution
The Fourier density-increment proof of the <Meshulam theorem> controls three-term progressions through ordinary Fourier coefficients, equivalently the $U^2$ norm. Four-term progressions are controlled by the <Gowers uniformity norm> $U^3$. A function can have small correlation with every linear character while correlating strongly with a <quadratic phase>; such a function can have small $U^2$ norm but large $U^3$ norm. Ordinary Fourier uniformity therefore does not make the four-term progression count pseudorandom, and a large linear Fourier coefficient need not exist when that count is deficient. Quadratic or higher-order Fourier structure is needed.
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