Let F(x) = (f₁(x), …, f_m(x)) be such that 1, f₁, …, f_m are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, de 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 ℝ^m with respect to the portion of the graph of F defined by a ≤ log åså ≤ T is at most O(1/(T — a)). We conclude that if I ⊆ 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 ∈ ℝ. It follows that the Borel chromatic number of the Cayley graph of ℝ^m with respect to the set {±F(s): s ∈ ℝ} is infinite. Analogous results are also proven when ℝ is replaced by the field of p-adic numbers ℚ_p. At the end, we will also show the existence of real analytic functions f₁, …, f_m, for which the analogous statements no longer hold.

令F（x）=（f ₁（x），…，f _m（x））使得1，f ₁，…，f _m是具有实系数的线性独立多项式。基于Bachoc，莲，奥利维拉和此Vallentin组合的想法与估计某些振荡积分与多项式阶段，我们将显示，ℝCayley图的独立性比^米相对于的图形的一部分˚F通过定义一个≤日志小号A≤ Ť至多Ô（1 /（T — a））。我们的结论是，如果我⊆ℝ^米具有正上部的密度，则该差集I -我包含表单的矢量˚F（小号）为无界的一组值小号∈ℝ。它遵循的ℝCayley图的波雷尔色数^米相对于集合{± ˚F（小号）：小号∈ℝ}是无限的。当ℝ由领域替换类似的结果也证明p进制数字ℚ _p。最后，我们还将展示真实解析函数f ₁，…，f _{m的存在。}，其类似的陈述不再适用。