On small densities defined without pseudorandomness

TL;DR

This paper introduces a weak assumption on linear forms over finite fields that ensures small density of certain subsets, nearly as weak as linear independence, with bounds close to optimal.

Contribution

It identifies a new, weaker condition than approximate joint equidistribution that guarantees small densities of specific subsets in finite fields.

Findings

The density of subsets avoiding full images under certain linear forms tends to zero as the number of forms increases.

The bound on the density is quasipolynomial in the number of forms, nearly optimal.

The assumption is almost as weak as linear independence, broader than previous conditions.

Abstract

We identify an assumption on linear forms $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \to \mathbb{F}_p$ that is much weaker than approximate joint equidistribution on the Boolean cube $\{0,1\}^n$ and is in a sense almost as weak as linear independence, but which guarantees that every subset of $\{0,1\}^n$ on which none of $\phi_1, \dots, \phi_k$ has full image has a density which tends to 0 with $k$. This density is at most quasipolynomially small in $k$, a bound that is necessarily close to sharp.