A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges.

There are a number of well-known and interesting results in this area. For example, the prime numbers are a large set, while the twin primes constitute a small set (Brun's theorem). Moreover, the Erdős–Turán conjecture is an example of an open problem connected to this.

A generalization of this could be done as follows. Define

$$S_f=\sum_{n=1}^\infty f(a_n)$$

where $a_n$ are the members of the set at hand and $f:\Bbb Z^+ \to \Bbb R$ is an arbitrary function (where $f(n) = \frac 1n$ in the usual small/large set definitions).

Now if $S_f < \infty$ the set could be said to be "small under $f(n)$", and "large under $f(n)$" if the sum diverges.

A not very interesting example is that $\{n^2, n \in \Bbb Z^+\}$ is "large under $f(n)=\frac 1{\sqrt n}$", but $\{n^3, n \in \Bbb Z^+\}$ would be small under the same conditions.

My question is if any work has been done with this (or possibly other) generalization(s) of the large/small set definition and if any interesting results have emerged from it.

  • $\begingroup$ This looks quite similar to what some authors call summable ideals. (See, for example, the book Farah I. Analytic Quotients.) I have also seen the notation $\mathcal I_c^{(q)}$ for the particular case $f(n)=n^{-q}$ in some papers. But this notation does not seem to be used a lot. $\endgroup$ – Martin Sleziak Aug 15 '14 at 12:40

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