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alanlu

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## 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/2

^{k}and b = (p + 1)/2

^{k}, 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.

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