Here is something which is bothering me a bit. You have rationals on a line. You can define a rational grid on R^n by taking points with all coordinates having rational values. Is there a generalization to generic finite dimensional manifolds?
There are three sets of properties I am particularly interested in.
- For every two rational numbers there exists an irrational number in between them and vice versa.
- The rationals are dense in R. The same for the irrationals.
- The rationals are a set of measure zero.
Property 2. is general enough that it doesn't need modification. 1. and 3. however does and this is how I would modify them.
1mod. For any two points p,q in S and any arc from p to q there exists a point on the arc in M-S. Also the same statement with S and M-S reversed.
3mod. For any point p on M and any chart U on M that contains p. The image of U intersection S is a set of measure 0.
My thought on a construct would be something like this. Chose some set of charts that cover M. Choose the charts so that the overlap maps maps rational grid onto rational grid. Then the union pull back of the rational grids on the images could be considered a "rational grid" of M.
The question is can such a .function be cleaned up and made rigorous?
PS. I seem to remember encountering something like this with zeta functions, but it's a vague memory.