In Reply to: hint for #2 posted by Chris#2 on June 03, 2003 at 00:03:21:
>i'm having some trouble rigorously "covering" (0,1)^n as you suggest in the hint. i know that you'd need at LEAST q^n boxes of (0,1/q)^n.
>ex: if i wanted to cover the 2-d square (0,1)^2 with (0,1/2) i'd need 4 of the smaller boxes (at least) which is 2*2=4.
>but how do i write this mathematically for the (0,1/q)^n case?
>i sure hope this isn't asking for too much of a hint.
>---chris#2
You'll need to write down a formula to describe a collection of
q^n boxes, each of which is a translate of (0,1/q)^n, which
are disjoint, and which are contained in (0,1)^n. (They won't
quite cover (0,1)^n perfectly; there will be some gaps because
the boxes are open. However, this will be good enough to obtain
the initial bound m( (0,1/q)^n ) <= 1/q^n).
For instance, when q=1/2 and n=2, the boxes (0,1/2)^2, (0,1/2) x (1/2,1), (1/2,1) x (0,1/2), and (1/2,1)^2 are all translates of
(0,1/2)^2 and are contained in (0,1)^2, and this can be used
(together with the properties of measure) to show that (0,1/2)^2
has measure at most 1/4.
Terry