|Kuratowski's Closure-Complement Cornucopia|
The motivation to build this site
can be traced back to early 1983 when I was a student in
D. A. Lind's undergraduate analysis course at the
University of Washington. When Cantor's ternary function was introduced,
a classmate named Peter asked about the values of its derivative on the Cantor set.
Lind sometimes gave us “lollipop” problems that provided an actual lollipop to the writer of the best solution. No lollipop was offered for answering this impromptu derivative question, but, seeming not to know the answer to it (he probably did though), Lind posed it as an interesting problem to investigate.
I was able to show that the derivative is positively infinite on a certain subset of the Cantor set. After seeing my work Professor Lind immediately posed a new question: with respect to the standard coin-flipping measure, does the Cantor function have infinite derivative almost everywhere on the Cantor set?
Since the Cantor set is the set of all points whose ternary representation contains no "1", its points can be randomized by flipping a coin an infinite number of times, writing "0" for heads and "2" for tails. Thus the question is, does this coin-flipping procedure almost always (with probability 1) produce the ternary representation of a point at which the derivative of the Cantor function is infinite?
This sounded very probabilistic to me. Having taken his probability course the previous year, I went to Ronald Pyke's office and showed him the question. After thinking on it briefly, Professor Pyke concluded that the answer is “yes” by the strong law of large numbers. He sketched out an argument and suggested that I write up a full proof to send to the American Mathematical Monthly's Problems and Solutions section.
This was all very exciting to me. However when I brought Professor Lind the news about proposing the problem to the Monthly, his response was muted.
This may be why:
This was Lind's first Monthly problem. He was still in high school when he proposed it. After receiving 79 solutions, the editors gave the problem its unflattering title.
If Professor Lind suspected that my proposal could easily be another well-known result thinly disguised, his suspicion was correct:
The problem turned out to follow immediately from a major theorem on derivatives!
Here is my second Monthly problem [1997 BRR PMark Bowron, Stanley Rabinowitz, Closure, Complement, and Arbitrary Union, Problem 10577, Amer. Math. Monthly, v. 104 no. 2, 1997, p. 169, Solution: John Rickard, v. 105 no. 3, 1998, pp. 282‑283.], co-proposed with Stanley Rabinowitz:
When we proposed it, neither of us knew that Kuratowski had already published a solution to this problem in 1922, with still more solutions appearing later!
The unoriginality of these two Monthly problems motivated me to publish the large list of references and English translations found on this website.
17 Jan 2015