Let $$D = \left\{\left(\frac{i}{p}, \frac{j}{p}\right): p \in \mathbb{N}, i,j = 1,2,…., p-1 \right\}$$
Determine whether $D$ has content $0$.
My thoughts:
First the definition of a set has (Jordan) content 0 given in the book is given below:
A set $D \subset \mathbb{R^2}$ has (Jordan) content 0 if for every $\epsilon > 0$ there exists a finite collection of rectangles $R_{k},$ $1 \leq k \leq n,$ whose union covers $D$ and the sum of their areas is less than $\epsilon.$
And I can see if I take an interval of the form $[-\epsilon/2p , \epsilon/2p]$, the given set $D$ will have a content 0. am I correct? if no then I do not know how to solve it, could anyone please give me a hint for the solution?
Best Answer
Your set $D$ coincides with $\bigl({\mathbb Q}\>\cap\ ]0,1[\,\bigr)^2$, because any such point $\bigl({p\over q},{r\over s}\bigr)$ can be written in the given form. You cannot cover this set with finitely many rectangles of total area $<1$: Any such configuration would leave some open part of $[0,1]^2$ uncovered, and this part would certainly contain points with rational coordinates.