= Quadratic uniformity of a set
{title2=$\|1_A-|A|/|G|\|_{U^3}\leq\alpha$}
= Quadratically uniform
{synonym}
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.
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