Polynomial configurations in sets of positive upper density over local fields

Abstract

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on 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 $\mathbb{R}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. We conclude that if $I \subseteq \mathbb{R}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ for an unbounded set of values $s \in \mathbb{R}$. It follows that the Borel chromatic number of the Cayley graph of $\mathbb{R}^m$ with respect to the set $\{ \pm F(s): s \in \mathbb{R} \}$ is infinite. Analogous results are also proven when $\mathbb{R}$ is replaced by the field of $p$-adic numbers $\mathbb{Q}_p$. At the end, we will also the existence of real analytic functions $f_1, \dots, f_m$, for which the analogous statements no longer hold.