Disjoint dyadic squares in unit disk

In summary: I still need to show that there must be at least one square in the collection that has a total area that exceeds π - ε.
  • #1
alanlu
65
0

Homework Statement



Given ε > 0, show that there is a collection of disjoint dyadic squares in the unit disk that has a total area which exceeds π - ε.

Homework Equations



Define a dyadic interval as an interval of the form [a, b] such that a = p/2k and b = (p + 1)/2k, p and k are integers. A dyadic square is the product of two of these intervals.

This problem is ch1 22b of Pugh's Real Mathematical Analysis.

The Attempt at a Solution



I figured this had something to do with aliasing on an circular arc, so I wrote [tex]A(n) = { 1 \over 4^n } \sum_{i=1}^{2^n} \lfloor \sqrt{4^n - i^2} \rfloor[/tex] as an expression where A is the area of a quadrant of the unit circle aliased down to a [itex]{ 1 \over 2^n }[/itex] granular dyadic grid. Then I noted that A → π/4 as n → ∞. What I am stuck on is the disjoint condition, as that makes requiring larger chunks of squares mandatory for the collection's total area to approach π I think, but it feels like to me that the supremum I want is less than π.
 
Last edited:
Physics news on Phys.org
  • #2
I don't understand your difficulty. The squares in A(n) are disjoint. What do you mean by "makes requiring larger chunks of squares mandatory"?
 
  • #3
I believe the squares include their boundaries, and by definition of the disjoint condition two adjacent squares would have a line/vertex (nonempty) intersection and so would not be disjoint. If I interpreted the question wrong and things are fine, then ... haha yeah, there isn't much difficulty after all. On the other hand, if the disjoint condition is that strict, then intuitively, I'm not even sure the question can be resolved.
 
  • #4
I don't think anyone's going to worry about line and point overlaps, which are clearly sets of measure zero. But if it bothers you, use intervals like [a, b) or (a, b).
 
  • #5
Alright, thank you haruspex. I'm starting to think the stricter interpretation has an interesting upper bound in its own right, but I don't think this would not be the forum for that discussion. :)
 
  • #6
I asked the author for some clarification and he has chosen to grace me with a response! Here's the quote:

Me: ...I asked online for some help and the advice given seems to indicate that perhaps the problem is worded too strictly...

Charles: ...Nope, the problem is correct as stated. Think of a pile of sand. If you can take away 1/16 of the pile, and then take 1/16 of the remaining pile, and continue doing that indefinitely, how much will be left in the limit?

In short, I'm not quite done yet.
 

Related to Disjoint dyadic squares in unit disk

1. What are disjoint dyadic squares in a unit disk?

Disjoint dyadic squares in a unit disk refer to a set of squares with sides parallel to the coordinate axes, where each square has a side length that is a power of two and the squares do not overlap.

2. How are disjoint dyadic squares related to unit disks?

Disjoint dyadic squares are typically studied in the context of unit disks, which are disks with a radius of 1 unit. These squares are often used to approximate unit disks, as they can be tiled to cover the entire disk.

3. What is the significance of studying disjoint dyadic squares in unit disks?

Disjoint dyadic squares in unit disks have many applications in areas such as geometric measure theory, harmonic analysis, and fractal geometry. They also have connections to topics such as Fourier series and the Riemann zeta function.

4. How are disjoint dyadic squares constructed in a unit disk?

Disjoint dyadic squares in a unit disk can be constructed using a process known as dyadic subdivision. This involves dividing the unit disk into smaller and smaller squares, where each square is divided into four equal subsquares.

5. What are some open questions regarding disjoint dyadic squares in unit disks?

While much is known about disjoint dyadic squares in unit disks, there are still some open questions in this area. Some of these questions include finding the optimal number of squares needed to cover a unit disk and determining the maximum overlap between two sets of disjoint dyadic squares.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
728
  • Calculus and Beyond Homework Help
Replies
18
Views
2K
Replies
2
Views
4K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
5K
Replies
1
Views
2K
  • Topology and Analysis
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Replies
1
Views
853
Back
Top