= Hyperplane density increment for cap sets
{title2=$\alpha'\geq\alpha+\alpha^2/2$}
If a <cap set> $A\subseteq\mathbb F_3^d$ has <subset density> $\alpha>0$ and $3^d\alpha^2\geq2$, some <affine subspace> of codimension one has relative <subset density> at least $\alpha+\alpha^2/2$. The zero-sum count and the <Parseval identity on a finite group> give a nonzero <finite abelian Fourier coefficient> of magnitude at least $\alpha^2/2$. The three slice densities are $\alpha+2\operatorname{Re}(z\omega^j)$, where $z$ is this <finite abelian Fourier coefficient> and $\omega$ a primitive cube <root of unity>; one slice has increment at least $|z|$. Translation preserves the zero-sum condition because the <characteristic> is three.
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