 Four-term progressions in a quadratically uniform interval set

Balance the indicator by its density on the interval, and extend it by zero to a cyclic group of order greater than twelve times the interval length and coprime to six. An eighth-power interval cube bound $\alpha$ bounds every four-term counting error by ${\alpha }^{1/8}$ through three Cauchy–Schwarz steps. The density contribution is bounded below by a constant times ${\delta }^4$, and the diagonal contribution tends to zero with interval length. A sufficiently small constant times ${\delta }^{32}$ therefore guarantees a nonconstant four-term progression. Stating the cube normalization distinguishes this parameter from the norm convention.