Large set (combinatorics)

For other uses of the term, see Small set (disambiguation).

In combinatorial mathematics, a large set of positive integers

S = \{s_0,s_1,s_2,s_3,\dots\}

is one such that the infinite sum of the reciprocals

\frac{1}{s_0}+\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}+\cdots

diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges.

Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions.

Examples

\{1, \dots, 6, 8, \dots, 16, 18, \dots, 66, 68, 69, 80, \dots \}
of integers whose decimal expansion does not include the digit 7 is small. Such series are called 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 involving large sets

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.

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.

See also

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.

References

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