The ordinary Cantor set is obtained by removing the middle third of a unit line segment and iterating. The resulting set of points is uncountable, nowhere dense and has zero measure.
It transpires that the construction can be modified to give a set with non-zero measure, the so-called fat Cantor set. Note that in the previous construction, we remove intervals whose total length sums to 1 (namely 1/3 + 2/9 + 4/27 + 8/81 + …). If, instead, we remove smaller intervals so that the sum converges to a value below 1, then the set left over will have non-zero measure. An example of this is the Smith-Volterra-Cantor set, with measure 1/2.
Indeed, by a similar construction, we can get nowhere dense sets with measure arbitrarily close to (but not equal to) 1. This result is so counter-intuitive that the fat Cantor set was proclaimed ‘bizarre mathematical object of the week’ by Ben Elliott et al.
It took me a while to see (it’s a while since I’ve been in school) that the Cantor set is indeed uncountable, In the end I settled on that if I take any number between 0 and 1 written in binary (0.10110101…), interpret it as ternary and multiply by two (0.20220202…) it lies on Cantor set. I hope it’s correct approach…
Extending this approach to other number bases, it looks like for every base, the set of all numbers interpreted as numbers in higher base have measure 0.
Yes, that’s the most natural bijection between the reals in [0,1) and the points in the Cantor set. Your generalisation is also correct. Quite an interesting related fact is that the harmonic series (1 + 1/2 + 1/3 + 1/4 + …) diverges, whereas the sum of the reciprocals of numbers with no ‘9’s in their decimal representation converges (to about 60, if I remember correctly).
Assuming 0.11111111… in binary, resp. 0.22222222… in ternary is equal to 1, it should work on [0,1] rather than on [0,1).
“the fat Cantor set was proclaimed ‘bizarre mathematical object of the week’ by Ben Elliott et al.”
Where? When I Google ‘”bizarre mathematical object of the week”’ or ‘”ben elliott” “fat cantor set”‘, the only result is this entry.
Which week was that? Are there other weeks?
There were indeed many other things named “bizarre mathematical object of the week”, but I can’t recall them all (another one is the function with a single stationary point which is a local but not global minimum; see the latest cp4space post). I’ll e-mail Ben Elliott to see if he has a record of them.
Ben Elliott here, confirming that I do not, in fact, have a record of them.
That’s a shame. Do you know whether Hunter Spink has such a record? (I seem to recall that our asymmetric Bettsian polychoron was BMOOTW for one of the last weeks of Michaelmas.)
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