Large set (combinatorics)

In combinatorial mathematics, a large set of positive integers
is one such that the infinite sum of the reciprocals
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

Every finite subset of the positive integers is small.
The set of all positive integers is a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression is a large set.
The set of square numbers is small. So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive integer values of any polynomial of degree 2 or larger forms a small set.
The set of powers of 2 is a small set, and so is any geometric progression.
The set of prime numbers is large. The set of twin primes is small.
The set of prime powers which are not prime is small although the primes are large. This property is frequently used in analytic number theory. More generally, the set of perfect powers is small; even the set of powerful numbers is small.
The set of numbers whose expansions in a given base exclude a given digit is small. For example, the set of integers whose decimal expansion does not include the digit 7 is small. Such series are called Kempner series.
Any set whose upper asymptotic density is nonzero, is large.
The set of all primes in an arithmetic progression, where a and b are coprime is large.

Properties

Every subset of a small set is small.
The union of finitely many small sets is small, because the sum of two convergent series is a convergent series.
The complement of every small set is large.
The Müntz–Szász theorem states that a set is large if and only if the set of polynomials spanned by is dense in the uniform norm topology of continuous functions on a closed interval in the positive real numbers. This is a generalization of the Stone–Weierstrass theorem.

Open problems involving large sets

Paul Erdős conjectured that all large sets contain arbitrarily long arithmetic progressions. He offered a prize of $3000 for a proof, more than for any of his other conjectures, and joked that this prize offer violated the minimum wage law. The 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.