Additional results to a theorem of Eisenträger and Everest
Let \(\mathcal{P}\) be the set of prime numbers and for any subset \(\mathcal{W}\subseteq \mathcal{P}\) let \[ \lim_{n\rightarrow +\infty} \frac{|\{p\leq n\,:\;p\in\mathcal{W}\}|}{\pi(n)} \] be the \textit{natural density} of \(\mathcal{W}\) (whenever the limit exists), and \[ \mathcal{O}_{\mathcal{W}} = \{x\in \mathbb{Q}\,:\;v_p(x)\geq 0\;\forall\,p\not\in \mathcal{W}\} \] be the ring of \(\mathcal{W}\)-integers (where \(v_p\) is the usual \(p\)-adic valuation). The paper deals with Hilbert's tenth problem over the rings \(\mathcal{O}_{\mathcal{W}}\,\), i.e., the possibility of finding an algorithm which decides whether a multivariable polynomial over \(\mathcal{O}_{\mathcal{W}}\) has a root in \(\mathcal{O}_{\mathcal{W}}\) (if such an algorithm does not exist, we say that the problem is \textit{undecidable} over \(\mathcal{O}_{\mathcal{W}}\)). The author builds on earlier work of \textit{B. Poonen} [J. Am. Math. Soc. 16, No.~4, 981--990 (2003; Zbl 1028.11077)] (which provides \(\mathcal{W}\) of natural density 1 such that the problem is undecidable over \(\mathcal{O}_{\mathcal{W}}\)) and \textit{K. Eisenträger} and \textit{G. Everest} [Proc. Am. Math. Soc. 137, No.~6, 1951--1959 (2009; Zbl 1267.11120)] (which provides a partition of \(\mathcal{P}\) in two sets \(\mathcal{V}\) and \(\mathcal{W}\) such that the problem is undecidable over both \(\mathcal{O}_{\mathcal{V}}\) and \(\mathcal{O}_{\mathcal{W}}\)) to show that one can choose \(\mathcal{V}\) and \(\mathcal{W}\) to have natural density \(r\) and \(1-r\) (respectively) for any \textit{computable} \(r\in[0,1]\) (where \(r\) is computable if there exists two sequences of integers \(\{a_n\,\}\) and \(\{b_n\,\}\) such that \(\frac{a_n}{b_n}\) converges to \(r\)). Moreover, refining the techniques of the papers mentioned above, the author is able to provide sets \(\mathcal{V}\) and \(\mathcal{W}\) which separate any given finite (and disjoint) sets of primes \(P_1\) and \(P_2\,\).
scientific article; zbMATH DE number 5944302