# Small set (combinatorics)

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

$S=\{s_{0},s_{1},s_{2},s_{3},\dots \}$ is one such that the infinite sum

${\frac {1}{s_{0}}}+{\frac {1}{s_{1}}}+{\frac {1}{s_{2}}}+{\frac {1}{s_{3}}}+\cdots$ 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
$\{\dots ,6,8,\dots ,16,18,\dots ,66,68,69,80,\dots \}$ is small. (This has been generalized to other bases as well.) See Kempner series.

## Properties

$\{1,x^{s_{1}},x^{s_{2}},x^{s_{3}},\dots \}\,$ 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. This question is still open.