Fourier uniformity on subspaces

TL;DR

This paper proves that for subsets of vector spaces over the finite field with two elements, the uniformity can be achieved on a subspace starting at zero, unlike over the field with three elements where this is not always possible.

Contribution

The paper establishes that in the case of the finite field with two elements, the uniformity subspace can be chosen to include the origin, which is not necessarily possible over the field with three elements.

Findings

Over _2, the uniformity subspace can be taken to start at zero.

Counterexample shows this is not always possible over _3.

A subsequent improvement by F. Manners provides better bounds for the main theorem.

Abstract

Let $\mathbb{F}$ be a fixed finite field, and let $A \subset \mathbb{F}^n$. It is a well-known fact that there is a subspace $V \leq \mathbb{F}^n$, $\mbox{codim} V \ll_{\delta} 1$, and an $x$, such that $A$ is $\delta$-uniform when restricted to $x + V$ (that is, all non-trivial Fourier coefficients of $A$ restricted to $x + V$ have magnitude at most $\delta$). We show that if $\mathbb{F} = \mathbb{F}_2$ then it is possible to take $x = 0$; that is, $A$ is $\delta$-uniform on a subspace $V \leq \mathbb{F}^n$. We give an example to show that this is not necessarily possible when $\mathbb{F} = \mathbb{F}_3$. ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners showed us a rather short argument he had found in 2013, giving a better bound for our main theorem. We do not, therefore, intend to publish this note. The example over $\mathbb{F}_3$ may still be of interest to some readers and so we will not withdraw the paper from the arxiv.