TheInfoList

In
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other are ...
mathematics, a large set of
positive integer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s :$S = \$ is one such that the
infinite sum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the reciprocals :$\frac+\frac+\frac+\frac+\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

* 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 #REDIRECT Harmonic series Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig ...
{{R from other capitalisation ...

. More generally, any
arithmetic progression An Arithmetic progression (AP) or arithmetic sequence is a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes an ...

(i.e., a set of all integers of the form ''an'' + ''b'' with ''a'' ≥ 1, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) is a large set. * The set of
square number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s is small (see
Basel problem The Basel problem is a problem in mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measu ...
). 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of degree 2 or larger forms a small set. * The set of powers of is known to be a small set, and so is any
geometric progression In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

(i.e., a set of numbers of the form of the form ''ab''''n'' with ''a'' ≥ 1, ''b'' ≥ 2 and ''n'' = 0, 1, 2, 3, ...). * The set of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s has been proven to be large. The set of
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pri ...
s has been proven to be small (see
Brun's constant In number theory, Brun's theorem states that the sum of the Multiplicative inverse, reciprocals of the twin primes (pairs of prime numbers which differ by 2) Convergent series, converges to a finite value known as Brun's constant, usually denoted b ...

). * The set of
prime power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s which are not prime (i.e., all numbers of the form ''p''''n'' with ''n'' ≥ 2 and ''p'' prime) is a small set although the primes are a large set. This property is frequently used in
analytic number theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. More generally, the set of
perfect power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s is small, even the set of
powerful number A powerful number is a positive integer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...
s is small. * The set of numbers whose expansions in a given
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, ...
exclude a given digit is small. For example, the set :$\$ :of integers whose
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
expansion does not include the digit 7 is small. Such series are called
Kempner series The Kempner series is a modification of the harmonic series (mathematics), harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum : \frac where the prime indicates that ''n' ...
. * The set of
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pri ...
s is small, but it is still conjectured that there are infinitely many twin primes. * Any set whose upper
asymptotic density In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number the ...
is nonzero, is large.

# Properties

* Every
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of a small set is small. * The union of finitely many small sets is small, because the sum of two
convergent series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is a convergent series. (In set theoretic terminology, the small sets form an
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
.) * The complement of every small set is large. * The Müntz–Szász theorem states that a set $S=\$ is large if and only if the set of polynomials spanned by ::$\$ :is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
in the
uniform norm In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathe ...
topology of
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s on a closed interval. This is a generalization of the
Stone–Weierstrass theorem In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemati ...
.

# Open problems involving large sets

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician In this page we keep the names in Hungarian order (family name first). {{compact ToC , short1, side=yes A * Alexits György (1899–1 ...

famously asked the question of whether any set that does not contain arbitrarily long
arithmetic progression An Arithmetic progression (AP) or arithmetic sequence is a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes an ...

s 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.
Carl Pomerance Carl Bernard Pomerance (born 1944 in Joplin, Missouri, Joplin, Missouri) is an American number theory, number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation provin ...

Paul Erdős, Number Theorist Extraordinaire
(Part of the article ''The Mathematics of Paul Erdős''), in ''
Notices of the AMS ''Notices of the American Mathematical Society'' is the membership journal A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organizations *Diary, a record of what happened over the ...
'',
January, 1998
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.

*
List of sums of reciprocals In mathematics and especially number theory, the sum of reciprocals generally is computed for the multiplicative inverse, reciprocals of some or all of the positive number, positive integers (counting numbers)—that is, it is generally the sum ...

# References

* A. D. Wadhwa (1975). An interesting subseries of the harmonic series. ''American Mathematical Monthly'' 82 (9) 931–933. {{JSTOR, 2318503 Combinatorics Integer sequences Mathematical series