Quadratic uniformity of a set ($\Vert 1_A-|A|/|G|{\Vert }_{U^3}\le \alpha$)

A subset $A$ of a finite abelian group $G$ with subset density $\eta =|A|/|G|$ is quadratically $\alpha$-uniform, in the norm convention, if its balanced subset indicator $g=1_A-\eta$ has Gowers U3 norm at most $\alpha$. This controls correlations on additive three-dimensional cubes after the subset density contribution is removed. If a convention instead bounds the eighth power of the norm by $\alpha$, its parameter corresponds to the norm parameter ${\alpha }^{1/8}$. The convention must therefore be stated when giving quantitative thresholds.

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.

For $A\subseteq {\mathbb{Z}}_N$, $N\ge 16$, put $\eta =|A|/N$, $M=\lfloor N/16\rfloor$, $J=\{0,\dots ,M-1\}$, and $w=1_J\ast 1_J$ using normalized convolution on a finite group. The weights $w(x)w(d-M)$ are supported where $0\le x\le 2M-2$ and $M\le d\le 3M-2$, so $x,x+d,x+2d,x+3d$ are distinct integer points below $N$. If $t=M/N$, then $\mathbb{E}w=t^2$ and the sum of absolute values of its Fourier coefficients is $t$, by Parseval identity on a finite group. Expanding the four indicators around their subset density leaves four terms with one balanced factor. Expand both weights in Fourier series and absorb each character $e(rx+sd)$ into the factors at $x$ and $x+d$. Character multiplication preserves the Gowers U3 norm, and the cyclic four-term progression bound with torsion factors bounds every error by $6^{1/8}\alpha$. Thus the weighted count is at least ${\eta }^4{t}^4-4\cdot 6^{1/8}\alpha t^2$. Since $t\ge 1/32$, positivity follows from $\alpha <{\eta }^4/(4096\cdot 6^{1/8})$.