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

  1. Feb 10, 2014 #1
    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?
     
    Last edited: Feb 10, 2014
  2. jcsd
  3. Feb 10, 2014 #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?
     
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