# Small set (combinatorics)

{{#invoke:Hatnote|hatnote}} In combinatorial mathematics, a small set of positive integers

${\displaystyle S=\{s_{0},s_{1},s_{2},s_{3},\dots \}}$

is one such that the infinite sum

${\displaystyle {\frac {1}{s_{0}}}+{\frac {1}{s_{1}}}+{\frac {1}{s_{2}}}+{\frac {1}{s_{3}}}+\cdots }$

converges. A large set is one whose sum of reciprocals diverges.

## Examples

• The set of numbers whose decimal representations exclude 7 (or any digit one prefers) is small. That is, for example, the set
${\displaystyle \{\dots ,6,8,\dots ,16,18,\dots ,66,68,69,80,\dots \}}$
is small. (This has been generalized to other bases as well.) See Kempner series.

## Properties

• A union of finitely many small sets is small, as the sum of two convergent series is a convergent series. A union of infinitely many small sets is either a small set (e.g. the sets of p2, p3, p4, ... where p is prime) or a large set (e.g. the sets ${\displaystyle \{n^{2}+k:n>0\}}$ for k > 0). Also, a large set minus a small set is still large. A large set minus a large set is either a small set (e.g. the set of all prime powers pn with n ≥ 1 minus the set of all primes) or a large set (e.g. the set of all positive integers minus the set of all positive even numbers). In set theoretic terminology, the small sets form an ideal.
${\displaystyle \{1,x^{s_{1}},x^{s_{2}},x^{s_{3}},\dots \}\,}$
is dense in the uniform norm topology of continuous functions on a closed interval. This is a generalization of the Stone–Weierstrass theorem.

## Open problems

It is not known how to identify whether a given set is large or small in general. As a result, there are many sets which are not known to be either large or small.

Paul Erdős famously asked the question of whether any set that does not contain arbitrarily long arithmetic progressions must necessarily be small. He offered a prize of \$3000 for the solution to this problem, more than for any of his other conjectures, and joked that this prize offer violated the minimum wage law.[1] This question is still open.

## Notes

1. Carl Pomerance, Paul Erdős, Number Theorist Extraordinaire. (Part of the article The Mathematics of Paul Erdős), in Notices of the AMS, January, 1998.