Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the rational numbers is used as the domain, it appears that the range is the quadratic irrationals. Can we generalize this relationship?
On the unit interval, call the dyadic rationals Box Set 0, the rationals Box Set 1, the quadratic irrationals Box Set 2, and in general, when Box Set K is the domain, call the range mapped by Conway's box function Box Set K+1.
Question 1:What is the rule for determining Box Set N for a given natural number N?
Since each Box Set is countably infinite (Aleph Null), and the real numbers on the unit interval are not countably infinite (at least Aleph One), there must be a set of the real numbers which will never be contained in any Box Set N as N goes to infinity. We may call that set the "unboxables".
Question 2: What is the "unboxable" set? (Or conversely, what is the "boxable" set: Box Set N as N goes to infinity?)
(Note also that applying the Question Mark function will also produce an infinite set of progressively "sparser" yet still countably infinite sets of rational numbers as Box Sets -1 through -N as N goes to infinity, raising related questions.)