Almost Affine Invariance Over Prime Fields: Green Problem 90

Abstract

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

14 pages. Comments are welcome