Polynomial patterns in subsets of large finite fields of low characteristic

Abstract

We prove a low characteristic counterpart to the main result in (Peluse, 2019), establishing power saving bounds for the polynomial Szemerédi theorem for certain families of polynomials. Namely, we show that if $P_1,\dots ,P_m\in ({\mathbb{F}}_p[t])[y]$ satisfy an equidistribution condition, which is a natural variant of the independence condition in (Peluse, 2019) for our context, then there exists $\gamma >0$ such that for any $q=p^k$ and any $A_0,A_1,\dots ,A_m\subseteq {\mathbb{F}}_q$, \begin{align*} \left| \left\{ (x,y) \in \mathbb{F}_q^2 : x \in A_0, x + P_1(y) \in A_1, \dots, x + P_m(y) \in A_m \right\} \right| = q^{-(m-1)} \prod_{i=0}^m{|A_i|} + O_{q \to \infty; P_1, \dots, P_m} \left( |A_0|^{1/2} q^{3/2 - γ} \right). \end{align*} In particular, if $A\subseteq {\mathbb{F}}_q$ contains no pattern $\{x,x+P_1(y),\dots ,x+P_m(y)\}$ of cardinality $m+1$, then \begin{align*} |A| \ll_{P_1, \dots, P_m} q^{1 - γ/ \left( m + \frac{1}{2} \right)}. \end{align*}

23 pages