Enumeration of all positive fractions recently has gained renewed interest (see the list below).

By translation invariance we can be sure that in all intervals (n, n+1] of the real axis, there are the same number of fractions: #(n, n+1] = #(m, m+1] for all natural numbers n and m.

That is a matter of symmetry, independent of the method used for counting them.

Cantor's famous sequence

1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...

however has half of its hits in the first unit interval and less than 1/n in the interval (n, n+1]. This raises two important questions: (1) What are the precise values of the function f(n) = #(n, n+1]/#(0, 1] in the limit? (2) how can the deviation from f(n) = 1 be interpreted?

None of these question could be answered in MathOverflow or Math.StackExchange.

My question: Can it be that this topic has never been treated in the literature or in lessons on set theory, for instance in the first years of set theory? I don't believe this but could not yet find any evidence.

List of sites

What fraction of fractions does Cantor's famous sequence enumerate? https://mathoverflow.net/questions/362791/what-fraction-of-fractions-does-cantors-famous-sequence-enumerate

Relative abundance of rationals in Cantor's bijection? https://math.stackexchange.com/questions/3708845/relative-abundance-of-rationals-in-cantors-bijection

Translation invariance https://groups.google.com/forum/#!topic/sci.logic/OIrleZcHXW0%5B1-25%5D

Enumerating the rationals https://groups.google.com/forum/#!topic/sci.math/D2UYGbt9Qh8

Translation invariance https://groups.google.com/forum/#!topic/scivszfc/tt_qy_ymAgo

Enumerations of the Rationals - two methods https://groups.google.com/forum/#!topic/sci.math/uudISZF3An4

  • $\begingroup$ Related: Robert Gray, Georg Cantor and transcendental numbers, American Mathematical Monthly 101 #9 (November 1994), 819-832. I would expect that anything since 1994 having to do with your questions would cite this paper because this paper is well known and possibly unique for the topic. However, the results of a google scholar search for papers that cite this paper didn't show anything I recognize as being very relevant. $\endgroup$ Jun 21, 2020 at 18:25
  • $\begingroup$ Sorry, I could not find anything related to this topic therein. $\endgroup$
    – Franz Kurz
    Jun 22, 2020 at 21:00
  • $\begingroup$ What do you mean by "the limit" here? Which limit? Cantor's enumeration of the rationals doesn't converge. $\endgroup$
    – Spencer
    Jun 27, 2020 at 16:02
  • $\begingroup$ The limit here means the complete enumeration. Take the first n indexes and then let n --> omega. $\endgroup$
    – Franz Kurz
    Jun 27, 2020 at 19:24
  • $\begingroup$ As far as I see Robert Grays article is not about asymptotic density of the enumeration in a selected interval, and hence irrelevant. $\endgroup$
    – user4900
    Jun 29, 2020 at 18:17


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