  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.
