# $\delta$-equidistributed polynomials over finite fields

I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $$p(\cdot)$$ from $$\mathbb{F}^n$$ to $$\mathbb{F}$$ is $$\delta$$-equidistributed is $$\forall c\in \mathbb{F}, |\{x\in \mathbb{F}^n : p(x)=c\}|=(1\pm \delta)|\mathbb{F}|^{n-1}$$.

I found a paper by Green and Tao that deals with it. They define the notion of rank of a polynomial: $$\operatorname{rank}_{d−1}(P)$$ is the smallest integer $$k$$ such that there exist degree $$d−1$$ polynomials $$q_1(X), ..., q_k(x)$$, and a function $$B : \mathbb{F}^k → \mathbb{F}$$, s.t. $$p(X) = B(q_1(X), ..., q_k(X))$$.

From theorem 1.7 in the paper, it follows that if a polynomial has high degree, it is equidistributed. However, I couldn't figure out the exact parameter/relationship between the equidistribution parameter $$\delta$$ and the rank needed to achieve that.

So my question is how high the rank should be (as a function of $$\delta$$) for a polynomial to be equidistributed?

• To my knowledge the most recent quantitative results in this direction are by Milicevic arxiv.org/abs/1902.09830 , though it turns out to be more convenient to use a Fourier-analytic notion of equidistribution (analytic rank) to get the sharpest results. – Terry Tao Oct 17 at 16:46