Today we finished off Chapter 11. We worked through some of the proofs, and discussing the meaning of some of theorems, and a good time was had by all.
In 11.10 it is mentioned that there are some notable constant multiples besides and in the bounding inequalities on approximating irrationals by rationals. However, the text doesn’t mention what they are, which I thought was unfortunate. I also wondered what sort of numbers are the troublesome examples for the constant . That is, the troublesome number for is the golden ratio (or anybody whose continued fraction ends in a string of 1s), so what numbers do it for . I think we decided that probably it was not a single number, but more like… any number whose continued fraction expansion is just lots of 1s and 2s. The more ones, the worse the number, in some sense. But as long as there are infinitely many twos, maybe you start running into, or getting close to, this bound.
We talked a little bit of our way through the proof that almost all numbers have arbitrarily large “quotients” (the in the continued fraction). I tried to dig up some memories from my reading of Khinchin’s book about how to picture some of the intervals and things in the proof. I have this pictures in my head of rectangles over the interval of height the length of the interval (I guess that makes them squares, huh?). So the biggest rectangle is the one between 1/2 and 1, and they get smaller as you move left. Then each rectangle is split up again, this time with the rectangles getting smaller as you move to the right (within one of the first-stage rectangles). The first set of rectangles correspond, somehow, to the first term of continued fractions, and the second (smaller) rectangles correspond to the second term. Probably I should dig out that book and try to figure out what this picture actually says, but for now… that’s the picture I have in my head.
We were all a little bit slow in understanding some of the later proofs about things like the discussion in 11.11: “Further theorems concerning approximations”. But we also didn’t seem interested enough to really dive in to it.
In the section on simultaneous approximations, Eric mentioned that similar things are done in other contexts (like, perhaps, -adics). When you have valuations, you prove a weak (single) and strong (simultaneous) theorem about approximations. While we were talking about it, I wondered if there was some analogy to the distinction between continuous (at each point in an interval) and uniformly continuous (on that interval). It seems like there maybe should be.
Finally, we spent a while digging through the proof that is transcendental. Mostly because I was stubbornly refusing to believe I wasn’t being lied to throughout the proof. Setting and then “plugging into” polynomials really made me uncomfortable. As we went, I joked about things not having any actual meaning. Eventually Chris and Eric pointed out that they do, actually, have meaning. This “plugging in” thing is actually giving you an integer (if your polynomial has integer coefficients). That calmed me down a bit. I still feel like I don’t understand the proof at all, and certainly couldn’t explain even an outline of it. Eric said similar things, but asked if we should have expected that somehow. Eric also mentioned that these sorts of formal manipulations with things that look wrong can sometimes be ok, and that it was something related to umbral calculus. He showed us an identity (Vandermonde’s) associated with binomial coefficients that does similar sorts of symbolic trickery. Which apparently I should now go read some more about.
I had printed out a paper about the continued fraction expansion of (which maybe was pointed out to me in this comment), which talked about Pade approximations. Some of the things looked somewhat similar to what was going on in the proof that is transcendental (which the paper said was where they came from), but I couldn’t explain the paper well during out meeting (since I don’t understand it well enough), and we ran out of time.