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[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk

I cam across this problem and I really don't know how to solve it.

So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the following picture :

enter image description here

Then you divide again each square in 4 new squares by the same process, so that you obtain the following picture and data :

enter image description here

Now put a circle inside the square :

enter image description here

If you repeat the process of dividing each square in 4, each new square having a new value, what is the value of the disk ?

I wrote a program allowing me to compute the value of the region outside the disk : I started with a square divided by $8 \times 8$ new squares and stopped at $2^{27} \times 2^{27}$.

Here is the output of the algorithm giving the approximation of the value of the disk (it is 1- approximation of the region outside the disk)

9.000000000e-01

8.144000000e-01

7.626000000e-01

7.292020000e-01

7.088800000e-01

6.973523000e-01

6.918611000e-01

6.885892690e-01

6.869197714e-01

6.859950674e-01

6.855135614e-01

6.852518648e-01

6.851172864e-01

6.850433926e-01

6.850051560e-01

6.849844363e-01

6.849737746e-01

6.849678775e-01

6.849649240e-01

6.849632929e-01

6.849624579e-01

6.849620047e-01

6.849617754e-01

6.849616479e-01

6.849615847e-01

I was not able to find an explicit formula for the limit (does it exist ?).

I also tried an exponential regression on the data but I was not really satisfied.

Any hint ?

[Added] I tried several configurations where I changed the parameters (values of the small squares) but I was not able to find anything...

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    $\begingroup$ If you expect your solution to be a simple linear expression in $\pi$, then $$\frac{3}{5}\pi-\frac{6}{5}\approx0.68495559$$ might be a reasonable guess. $\endgroup$
    – Moritz Firsching
    Aug 19, 2016 at 11:43
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    $\begingroup$ The Fourier transform of the measure on the square can be written as an infinite product whose terms are Fourier transforms of weighted sums of $4$ delta functions, all rescaled copies of the original. The Fourier transform of the uniform measure on the disk is known and related to a Bessel function. The integral of the product of these gives an expression for the measure inside the circle. $\endgroup$
    – Douglas Zare
    Aug 19, 2016 at 13:28
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    $\begingroup$ I used the LLL implementation of pari/gp: lindep([.684961,1,Pi],flag=4) gives [-5, -6, 3]. $\endgroup$
    – Moritz Firsching
    Aug 19, 2016 at 18:55
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    $\begingroup$ The sum over the square would just be 1 right? The unnaturalness of the circle is what is making this problem difficult it seems. $\endgroup$
    – Vigod
    Aug 20, 2016 at 3:59
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    $\begingroup$ Yes, the infinite product at a point is easy to write down. If you write the coordinates of the point in base 2, the digits tell you if you have to multiply by 0.1, 0.2, 0.3 or 0.4. The problem is to sum all these infinite products inside the circle. The constraint that you have to impose is unnatural since the circle is breaking the scaling symmetry of the problem. $\endgroup$
    – Vigod
    Aug 20, 2016 at 19:41

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