🚀 go-pugleaf

After a few days for this to soak into my head, I have some comments.

The "pixel counting" code is computing a Riemann sum.  The function
in question is 1 on the Cantor-like set, and 0 elsewhere.  Now this
set has empty interior and positive measure.  This means that the
function is not Riemann integrable.  So it would not be a surprise
if Riemann sums fail to converge to the integral--that is, pixel
counting fails to converge to the measure of the set.

There is a modified Riemann sum scheme that will converge to the Lebesgue
integral, even when the function is not Riemann integrable.  Still
subdivide the interval into many short intervals.  But don't use
the left-hand endpoint of each of them: instead use a randomly
chosen point of each interval.  Determine whether your point is in
the set or outside the set, accordingly count the point as "in" or
"out", and continue as before.  This will converge (with probability
one) to the Lebesgue measure.

By a random choice of a point in the interval, I do not mean merely
a rational number with a large power of 2 as its denominator.  I mean
a random real number (which will be irrational...).  Hope for computation with
such a number is not futile, however.  You would use your random number
generator to compute successive bits of the number.  You only have to
generate bits until you know whether your number is in the set.
As with the Mandelbrot set iteration, after a while you give up and say
"in" even if it is uncertain.

It might be interesting to implement this in Yuval Fisher's program and
see what happens.
-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)