 Hyperplane density increment for cap sets

If a cap set $A\subseteq {\mathbb{F}}_3^d$ has subset density $\alpha >0$ and $3^d{\alpha }^2\ge 2$, 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 $\alpha +2\mathrm{Re}(z{\omega }^j)$, 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.