Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:



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> $(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s$ such that for all $x\in [0,1]^n$ there is $y\in S$ such that $\| x-y \|< \frac{p}{q}$.
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**Question.** Is $R\subseteq \mathbb{N}^4$ computable?



Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls
The question whether $(p,q,n,s)\in R$ in any instance can be expressed as a sentence in the language of the structure $\langle\mathbb{R},+,\cdot,0,1,<\rangle$, a real-closed field, and by Tarski's theorem on real closed fields, there is a computable uniform decision procedure to decide the truth of all such sentences.


See further description at [my answer to a related question](https://mathoverflow.net/a/134260/1946).


