# Has Cantor's irregular enumeration of rationals ever been discussed?

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