Polynomial configurations in sets of positive upper density over local fields

Mohammad Bardestani, Keivan Mallahi-Karai

Published 2017-01-21Version 1

Let $\mathbb{K}$ be the field of real or $p$-adic numbers, and $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with coefficients in $\mathbb{K}$. Employing ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{K}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. From here, we conclude that if $I \subseteq \mathbb{K}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ for an upbounded set of values $s \in \mathbb{K}$. We deduce that the Borel chromatic number of the Cayley graph of $\mathbb{K}^m$ with respect to the set $\{ \pm F(s): s \in \mathbb{K} \}$ is infinite, while the clique number of this Cayley graph is finite. Moreover, the infinitness of the Borel chromatic number does not hold necessarily if $f_1, \dots, f_m$ are merely real-analytic functions.