On Monday, April 7, 2014 8:44:14 PM UTC+1, dull...@sprynet.com wrote: ...> HAve you got any idea what Cantor was working on, that _forced_ him to > > study infinite ordinals? I didn't think so. It was a problem in > > Fourier series! > > > > (Specifically, which sets E have the property that if a trig series > > converges to 0 on the complement of E then the coefficients > > vanish...) > > ...
I had no idea about this either, and I find this side-topic far more interesting than the actual thread.
From a few minutes casual googling, it seems that Cantor proved using transfinite induction (the origins for Cantor's development of set theory) that finite sets E have the above property. Others than proved that countable sets E have the same property.
Apparently, not all sets of Lebesgue measure 0 are sets of uniqueness. (A set of uniqueness is a set of the type described by E).
So this is some interesting classical mathematics. As you may know, from other threads, I like to read proofs of great classical theorems and then ask sci.math when I need help.
Many of my questions go unanswered (for example, when I got stuck on Szemeredi's proof of his theorem about subsets of the natural numbers of positive upper density).
I'm wondering what I can do, to improve my success at getting my questions answered. You (David Ullrich) made the excellent point that it's a good idea to learn a lot of the underlying theory first by doing related exercises. Maybe, that's really the only way to make progress, I don't know. With the Szemeredi proof, the original 1975 proof is so totally elementary that it's hard to see how doing other exercises can help or what those exercises would be.
A digression, I know, but perhaps that's forgivable.