A subset $S$ of $\mathbb{R}^n$ is "bounded away from integer points" if for some positive $\epsilon$ every point in $S$ lies at a distance of at least $\epsilon$ from $\mathbb{Z}^n$. For example the line $x+y=1/2$ in $\mathbb{R}^2$ is bounded away from integer points, but the curve $x^2+y=1/2$ is not, because the points $(n+\frac{1}{4n},-n^2-\frac{1}{16n^2})$ for $n=1,2,\ldots$ lie on this curve.

Question: Can anyone give an algorithm to determine whether a system of polynomial equations with real algebraic coefficients cuts out a subset $S$ of $\mathbb{R}^n$ that is bounded away from integer points? Is there a simple description of all such subsets $S$?

Remark: I have a vague notion that if $S$ is bounded away from integer points then this must be ``trivially'' verifiable, perhaps because $S$ projects on a linear affine subset of $\mathbb{R}^n$ that is obviously bounded away from integer points, but this is little more than guesswork. I don't actually know that the problem is decidable, but I would be surprised if it were not.

UPDATE: (I'll use this section to collect my latest thoughts on the problem.)

The situation is fairly transparent in $\mathbb{R}^2$, and the real problem is how things generalize to higher dimensions. Let $S$ be as above. Let $\lfloor\cdot\rfloor$ be the floor function, which will be applied to points coordinatewise. Then I propose the following conjecture:

There exists some translate of $S$ bounded away from integer points if and only if the set of all points $\lfloor p\rfloor$ for $p\in S$ is contained in a finite union of linear-affine subspaces of $\mathbb{R}^n$ (which will be defined over the rationals).