Large set (combinatorics)

Examples

• Every finite subset of the positive integers is small.
• The set of all positive integers is known to be 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 known to be a small set, and so is any geometric progression.
• The set of prime numbers has been proven to be large. The set of twin primes has been proven to be small.
• The set of prime powers which are not prime is a small set although the primes are a large set. This property is frequently used in analytic number theory. More generally, the set of perfect powers is small.
• The set of numbers whose expansions in a given base exclude a given digit is small. For example, the set
• Any set whose upper asymptotic density is nonzero, 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

  Open problems involving large sets