Well, somewhat surprisingly, my little reading group hasn’t yet disbanded entirely. Last week we had a great discussion about chapter 2 of H&W, which put us in chapter 3 for this week.
Chris commented that one of the more interesting things, for him, in this chapter was that people would actually be interested in proving things about sequences of reduced fractions with bounded denominator (the Farey sequences that start the chapter). I rather enjoyed playing with them and reading about them. My understanding is that they are useful for starting to get bounds on approximating irrationals by rationals, a topic which I’m fairly certain we will return to. I also was glad to see a proof of Minkowski’s theorem about symmetric regions with large area containing lattice points. I think I’d heard the result before, but it’s always nice to see (and be able to follow!) a proof.
I was a little confused about when a mediant is reduced. I thought that the mediant of reduced fractions was reduced. For any three consecutive terms of a Farey sequence, the middle is the mediant of the other two. There is a footnote stating that the middle term might be the reduced form of the mediant. Why would you need this footnote, if mediants are already reduced? Chris pointed out that in the Farey sequence with denominator no larger than 3 (denote if ), 1/3, 1/2, 2/3 are consecutive terms, but the middle is the reduced mediant of the outer terms. All the text claims is that the first time a term shows up, it is the reduced mediant of its neighbors. Clearly mediants of reduced fractions aren’t, necessarily, reduced. Eric and I were frequently embarrased by things we thought were true during this meeting, and I pointed out that this was exactly why I wanted to read about these things: I don’t work with integers too much
Eric wondered about the fact that the distance between two consecutive terms and in a Farey sequence is . Chris and I noted that this was obvious from the lemma that . Eric was wondering, though, if one could determine it without that lemma somehow. We didn’t come up with much.
I presented the answer I had found to the question I had asked myself: “For in , can you find the first for which has a new successor in ? What can you say about the sequence of all such ?” The book has a lemma about how to find the next term for in . Suppose that its successor is . By other lemmas in this chapter, the next time something comes between these terms, it will be the mediant, , and this occurs in . The next time will have a new successor, it will be the mediant of with this new term, , and so will be . Continuing on, we see that has a new successor in the Farey sequences .
Next, I talked about the Stern-Brocot Tree and Minkowski’s Question Mark function, . We had mentioned in our meeting that the terms of the Farey sequence are not equally distributed, and I pointed out that there was some relation between that and how ‘wobbly’ is. Notice that the Farey sequences are least dense around 1/2, and most dense around 0 and 1, which relates well to the wobbliness of the graph of . This has something to do with defining an appropriate measure so that is the integral (of… something. 1? ?) with respect to this measure. At least, that’s what I gathered from Wikipedia. I’d like to read more about this.
Also, Eric and I had both noticed the comment on the Wikipedia page that there is a relation between Farey sequences and the Riemann hypothesis, which we found pretty intriguing. Of course, neither of us knew much about it. Perhaps a topic for another day. It seems to be related to the density, or perhaps distribution is more accurate, of the terms of Farey sequences in the interval .
I completely forgot to bring up Ford circles. If you put circles of appropriate sizes above the points in a Farey sequence, you get lots of nice tangent circles, with fun properties. Perhaps the property most relevant to this section is that any circles tangent to the circle above are centered at -coordinates that are neighbors of in some Farey sequence.