Approximately Symmetric Forms Far From Being Exactly Symmetric

TL;DR

This paper investigates the structure of approximately symmetric multilinear forms over finite fields, disproving a conjecture that such forms are always close to symmetric forms, and providing explicit counterexamples with large differences.

Contribution

The paper demonstrates that d-approximately symmetric multilinear forms can be far from symmetric forms, countering previous conjectures and constructing explicit high-rank examples.

Findings

Counterexample shows large difference from symmetric forms

Disproves conjecture relating approximate symmetry to proximity

Constructs explicit forms with high partition rank difference

Abstract

Let $V$ be a finite-dimensional vector space over $\mathbb{F}_p$. We say that a multilinear form $\alpha \colon V^k \to \mathbb{F}_p$ in $k$ variables is $d$-approximately symmetric if the partition rank of difference $\alpha(x_1, \dots, x_k) - \alpha(x_{\pi(1)}, \dots, x_{\pi(k)})$ is at most $d$ for every permutation $\pi \in \operatorname{Sym}_k$. In a work concerning the inverse theorem for the Gowers uniformity $\|\cdot\|_{\mathsf{U}^4}$ norm in the case of low characteristic, Tidor conjectured that any $d$-approximately symmetric multilinear form $\alpha \colon V^k \to \mathbb{F}_p$ differs from a symmetric multilinear form by a multilinear form of partition rank at most $O_{p,k,d}(1)$ and proved this conjecture in the case of trilinear forms. In this paper, somewhat surprisingly, we show that this conjecture is false. In fact, we show that approximately symmetric forms can be quite far from the symmetric ones, by constructing a multilinear form $\alpha \colon \mathbb{F}_2^n \times \mathbb{F}_2^n \times \mathbb{F}_2^n \times \mathbb{F}_2^n \to \mathbb{F}_2$ which is 3-approximately symmetric, while the difference between $\alpha$ and any symmetric multilinear form is of partition rank at least $\Omega(\sqrt[3]{n})$.