Large set (combinatorics)

For other uses of the term, see small set.

In combinatorics, a small set of positive integers

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

is one such that the infinite sum

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

converges. A large set is any other set of positive integers.

For example, the set $\{1,2,3,4,5,\dots \}$ of all positive integers is known to be a large set (see Harmonic series (mathematics)), and the set $\{1,2,4,8,\dots \}$ of powers of 2 is known to be a small set. There are many sets about which it is not known whether they are large or small.

A union of small sets is small, as the sum of two convergent series is a convergent series. Also, a large set minus a small set is still large.

The set of prime numbers has been proven to be large.

The Müntz-Szasz theorem is that a set $S=\{s_{1},s_{2},s_{3},\dots \}$ is large iff

$\{1,x^{s_{1}},x^{s_{2}},x^{s_{3}},\dots \}$

is dense in the uniform norm topology of continuous functions on a closed interval. This is a generalization of the Weierstrass approximation theorem.

Another known fact is that 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.)

References

• A. D. Wadhwa (1975). An interesting subseries of the harmonic series. American Mathematical Monthly 82 (9) 931–933.

See also

• Harmonic series (mathematics)
• Series (mathematics)