1. Computability

Problem 1.1.
[Russell Miller] Given $f\in \mathbb Z[x_1,\dots, x_n]$, define the following sets. $$ A(f) = \{\text{subrings of }\mathbb Q \text{ where $f$ has a solution}\} $$ $$ \mathcal C(f) = \text{The interior of the complement of $A(f)$} $$ $$ B(f) =\text{The boundary of the complement of $A(f)$} %= \text{``the boundary set"} $$ Can the Lebesgue measure of $B(f)$ be positive?
 If so, what is the maximal complexity of the measure of $B(f)$?
 Is $\mathcal C(f)$ always a finite union of basic open sets?
Cite this as: AimPL: Definability and decidability problems in number theory, available at http://aimpl.org/definedecide.