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arXiv · 1902.00353

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture

James Aaronson
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Abstract

Let $p$ be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p$ can be stated as follows. If $ϕ: \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ is a function such that $ϕ(x+y) - ϕ(x) - ϕ(y)$ takes values in some set $S$, then there is a linear map $\tildeϕ : \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ with the property that $ϕ- \tildeϕ$ takes at most $|S|^{O(1)}$ values. A strong variant of this conjecture states that, in fact, there is a linear map $\tildeϕ$ such that $ϕ- \tildeϕ$ takes values in $tS$ for some constant $t$. In this note, we discuss a counterexample to this conjecture.

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James Aaronson. 2019-02-01. A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture. https://arxiv.org/abs/1902.00353

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