Almost Affine Invariance Over Prime Fields: Green Problem 90

TL;DR

This paper characterizes the threshold for sets in finite fields to be almost affine invariant under a family of affine transformations, solving an open problem posed by Ben Green.

Contribution

It determines the precise growth rate of the parameter K for which sets are almost affine invariant under all transformations with bounded coefficients, resolving Green's Open Problem 90.

Findings

Sets with density 1/2 are almost affine invariant if K=o(log p).

The threshold for invariance under affine transformations is K=o(log p).

The result solves an open problem by Ben Green.

Abstract

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 \[|A \triangle \phi(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $\phi(x)$ with $|a|, |b|\le K$ and $a\neq 0$, is $K=o(\log p)$. This solves Ben Green's Open Problem 90.