 Solution

Use ${\Delta }_{h}f(x)=f(x+h)\overline{f(x)}$ and the unnormalized Fourier transformWe state the eighth-power convention explicitly. A bounded function is $\alpha$-quadratically uniform ifwhere $\mathcal{C}$ means conjugation and every expectation is uniform. The expression is real and nonnegative. Applying Parseval identity on a finite group to the multiplicative derivatives gives the equivalent conditionThis is the Derivative identity for the Gowers U3 norm. A set $A$ of density $\delta$ is quadratically uniform when its balanced subset indicator $1_A-\delta$ satisfies that condition, not when its unbalanced indicator does. Some definitions bound $\Vert f{\Vert }_{U^3}$ rather than its eighth power; that norm parameter is ${\alpha }^{1/8}$ in the convention here. Quadratic uniformity controls three-dimensional additive cubes and four-term progression counts; small ordinary linear Fourier coefficients alone are not the same condition.