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Given any interval $I=(a,b) \subset \mathbb R^+$, we may order the rationals in $I$ with a denominator-first lexicographic order, as follows: First, we list, in increasing order of numerator, all $q \in I$ that have a denominator of $1$ when written in lowest terms (if any); then we list all $q \in I$ that have a denominator of $2$ when written in lowest terms (if any); and so forth. For example, the rationals in $(5,6)$ would be enumerated as: $$\frac{11}{2}, \frac{16}{3}, \frac{17}{3}, \frac{21}{4}, \frac{23}{4}, \frac{26}{5}, \frac{27}{5}, \frac{28}{5}, \frac{29}{5}, \frac{31}{6}, \frac{35}{6} \dots$$ This is obviously a well-ordering, and is similar in spirit to the Farey sequences of order $n$, although they are not the same thing. (In particular the order I am considering here is not compatible with the usual order $<$ on $\mathbb Q$.)

It is known, for example, that if $a,b$ are irrational then the "first" rational in $(a,b)$ (in the sense of this order) can be computed by comparing the continued fraction representations of $a$ and $b$.

My questions:

  1. Does this order have a standard name? "Denominator-first lexicographic order" is precise, but clumsy. Something non-descriptive but with a person's name attached to it (in the flavor of "Farey sequence") would be fine.
  2. What else (if anything) is known about this order? For example, is there a way to compute the second (or nth) rational in $(a,b)$ using continued fractions or something else?

If I knew a good answer to the first question, I suspect I could probably make some headway on my own with the second question. (Without a distinguished name, this turns out to be very hard to Google.)

For those who are interested in the background to this question, I am working on a paper about locating the first rational square in an interval $(a,a+1)$ (where "first" is relative to this order). I've found lots of interesting results but have been unsuccessful at finding related prior work to make connections to.

Edited to add: With less than 8 hours to go on the bounty, I thought I might provide some more details about the kind of directions I am interested in. Tad's answer below suggests looking at the "canonical" interval $(0,1)$, but what about intervals of the form $(\sqrt{n},\sqrt{n+1})$? Consider by way of example $(\sqrt{13},\sqrt{14})$. The rationals in this interval are, in this well-ordering, $$\frac{11}3, \frac{26}7, \frac{29}8, \frac{37}{10}, \frac{40}{11}, \frac{41}{11}, \frac{47}{13}, \frac{48}{13},\dots $$

Some things are easy to see: For example, the sequence of denominators $$(d_n): 3, 7, 8, 10, 11, 11, 13, 13, \dots$$ may have both skips and repeats. As noted above, the first term in the sequence (namely $\frac{11}3$) can be computed by comparing the continued fractions for $\sqrt{13}$ and $\sqrt{14}$. Is there an analogous way to compute the second or nth term in the sequence? Are there any other known properties of these sequences?

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The "canonical" interval to try is $(0,1)$. Based on that, go to the Online Encyclopedia of Integer Sequences and look up the sequence of numerators (which is the concatenation, for all $k\ge2$, of the $\phi(k)$ integers less than $k$ which are relatively prime to $k$: $1, 1, 2, 1, 3, 1, 2, 3, 4, 1, 5,\ldots$) Your sequence appears as sequence A038566; there are a ton of references there.

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    $\begingroup$ Thank you for the suggestion. Do you have any ideas how to deal with intervals that are not of the form $(n,n+1)$? For example, it isn't clear to me at all how to adapt the "canonical" interval to the case of one of the form $(\sqrt{n},\sqrt{n+1})$. $\endgroup$ – mweiss Jun 7 '15 at 18:05
  • $\begingroup$ The same approach works for any interval $(a,b)$ (list in order of increasing denominators, then increasing numerators). For a given denominator $q$ you need to find the least $p$ such that $p/q> a$, which is $aq+1$ if $a q\in \Bbb Z$ and $\lceil a q\rceil$ otherwise. However, while there's a chance there's a formula for the $n$-th rational in the list when $(a,b)=(0,1)$, there's not much chance when $(a,b)=(e,\pi)$. $\endgroup$ – Tad Jun 7 '15 at 22:13
  • $\begingroup$ Yes, I know that one can list the rationals that way for any interval -- that was in the original question. I am wondering if that ordering has a name, and what (if anything) is known about how that ordering behaves. $\endgroup$ – mweiss Jun 7 '15 at 22:42

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