Almost Affine Invariance Over Prime Fields: Green Problem 90

Let $A\subset {\mathbb{F}}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $\phi (x)=ax+b$ if $$\mathrm{\mid }A\mathrm{△}\phi (A)\mathrm{\mid }=o(p)\mathrm{.}$$ We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $\phi (x)$ with $\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid }\le K$ and $a\mathrm{\ne }0$, is $K=o(\log p)$. This solves Ben Green’s Open Problem 90.