How to interpret this problem involving dyadic squares and unit disc?

If that is the case, then the answer would be yes. In summary, the problem from "Real Mathematical Analysis" is asking whether there exists a finite set of dyadic squares in the unit disc whose total area exceeds ##\pi - \epsilon## and only intersect along their boundaries. The answer to this question is yes.
  • #1
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The following problem is from the textbook "Real Mathematical Analysis" by Charles Chapman Pugh.

Given ##\epsilon > 0##, show that the unit disc contains finitely many dyadic squares whose total area exceeds ##\pi - \epsilon##, and which intersect each other only along their boundaries.​

I find myself unable to understand what the problem is asking to show. At first glace, I thought it was asking to show that given any ##\epsilon##, any collection of dyadic squares that 1) had a total area of greater than ##\pi - \epsilon## and 2) intersected along their boundaries would necessarily be finite. However, this assertion is obviously not true. [Let ##\epsilon > \pi## and consider the infinite collection of dyadic squares ##\{[0,\frac{1}{2}]\times[0,\frac{1}{2}],[\frac{1}{2},\frac{3}{4}]\times[0,\frac{1}{4}],[\frac{3}{4},\frac{7}{8}]\times[0,\frac{1}{8}],...\}##]

What am I missing?
 
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  • #2
After some additional thought, maybe the problem is actually just asking if given an ##\epsilon > 0##, does there exists a finite set of dyadic squares whose area is greater than ##\pi - \epsilon##, and which intersect only along their boundaries?
 

Related to How to interpret this problem involving dyadic squares and unit disc?

1. What is the difference between a dyadic square and a unit disc?

A dyadic square is a square that can be divided into smaller squares with sides that are half the length of the original square. A unit disc, on the other hand, is a circle with a radius of 1 unit.

2. How do I determine the relationship between a dyadic square and a unit disc?

The relationship between a dyadic square and a unit disc depends on their relative sizes. If the diameter of the unit disc is equal to the side length of the dyadic square, then the disc will be tangent to the square. If the diameter is larger, then the disc will intersect the square. If the diameter is smaller, then the disc will be contained within the square.

3. What is the significance of dyadic squares and unit discs in this problem?

Dyadic squares and unit discs are commonly used in mathematics to represent geometric shapes and to solve problems involving geometric relationships. In this particular problem, they may be used to determine the intersection or containment of two shapes.

4. How can I interpret the problem using dyadic squares and unit discs?

To interpret the problem, you can start by drawing a dyadic square and a unit disc and labeling their dimensions. Then, you can use geometric principles and formulas to determine their relationship and solve the problem. It may also be helpful to break down the problem into smaller parts to better understand the relationships between the shapes.

5. Are there any other mathematical concepts or tools that can help me interpret this problem?

Yes, there are other mathematical concepts and tools that can be used to interpret this problem. Some examples include geometric theorems, trigonometry, and coordinate geometry. It is important to carefully read and analyze the problem to determine which concepts and tools are most relevant to solving it.

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