Equidistribution of high-rank polynomials with variables restricted to subsets of $\mathbb{F}_p$

TL;DR

This paper extends Green and Tao's results by showing that high-rank polynomials over finite fields exhibit equidistribution even when variables are restricted to subsets, using tensor rank properties.

Contribution

It generalizes equidistribution results to polynomials restricted to subsets of finite fields, linking tensor rank to polynomial distribution properties.

Findings

High-rank polynomials are equidistributed over subsets of finite fields.

Existence of a polynomial vanishing on the subset that relates to the original polynomial.

Bounded rank of the difference polynomial indicates equidistribution.

Abstract

Let $p$ be a prime and let $S$ be a non-empty subset of $\mathbb{F}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ that vanishes on $S^n$, such that the polynomial $P-P_0$ has bounded rank. Our argument uses two black boxes: that a tensor with high partition rank has high analytic rank and that a tensor with high essential partition rank has high disjoint partition rank.