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| {{For|other uses of the term|Small set (disambiguation)}}
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| In [[combinatorics|combinatorial]] mathematics, a '''small set''' of [[positive integer]]s
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| :<math>S = \{s_0,s_1,s_2,s_3,\dots\}</math>
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| is one such that the [[infinite sum]] | |
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| :<math>\frac{1}{s_0}+\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}+\cdots</math>
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| [[Series (mathematics)|converges]]. A '''large set''' is one whose sum of reciprocals [[Series (mathematics)|diverges]].
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| ==Examples==
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| * The set <math>\{1,2,3,4,5,\dots\}</math> of all positive integers is known to be a large set (see [[Harmonic series (mathematics)|Harmonic series]]), and so is the set obtained from any [[arithmetic sequence]] (i.e. of the form ''an'' + ''b'' with ''a'' ≥ 0, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) where ''a'' = 0, ''b'' = 1 give the multiset <math>\{1,1,1,\dots\}</math> and ''a'' = 1, ''b'' = 1 give <math>\{1,2,3,4,5,\dots\}</math>.
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| * The set of [[square number]]s is small (see [[Basel problem#The Riemann zeta function|Basel problem]]). So is the set of [[cube number]]s, the set of 4th powers, and so on. More generally, the set of values of a polynomial <math>a_k n^k+a_{k-1} n^{k-1}+\cdots+a_2 n^2+a_1 n+a_0</math>, ''k'' ≥ 2, ''a''<sub>''i''</sub> ≥ 0 for all ''i'' ≥ 1, ''a''<sub>''k''</sub> > 0. When ''k''=1 we get an arithmetic sequence (which forms a large set.).
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| * The set <math>\{1,2,4,8,\dots\}</math> of powers of [[2 (number)|2]] is known to be a small set, and so is the set of any [[geometric sequence]] (i.e. of the form ''ab''<sup>''n''</sub> with ''a'' ≥ 1, ''b'' ≥ 2 and ''n'' = 0, 1, 2, 3, ...).
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| * The set of [[prime number]]s [[Proof that the sum of the reciprocals of the primes diverges|has been proven]] to be large. The set of [[twin prime]]s has been proven to be small (see [[Brun's constant]]).
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| * The set of [[prime power]]s which are not prime (i.e. all ''p''<sup>''n''</sup> with ''n'' ≥ 2) 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 power]]s is small.
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| * The set of numbers whose [[decimal]] representations exclude ''7'' (or any digit one prefers) is small. That is, for example, the set
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| :<math>\{\dots, 6, 8, \dots, 16, 18, \dots, 66, 68, 69, 80, \dots \}</math>
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| : is small. (This has been generalized to other [[Numeral system|bases]] as well.) See [[Kempner series]].
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| ==Properties==
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| * A [[union (set theory)|union]] of finitely many small sets is small, as the sum of two [[convergent series]] is a convergent series. A union of infinitely many small sets is either a small set (e.g. the sets of ''p''<sup>2</sup>, ''p''<sup>3</sup>, ''p''<sup>4</sup>, ... where ''p'' is prime) or a large set (e.g. the sets <math>\{n^2 + k : n > 0 \}</math> for ''k'' > 0). Also, a large set [[complement (set theory)|minus]] a small set is still large. A large set minus a large set is either a small set (e.g. the set of all prime powers ''p''<sup>''n''</sup> with ''n'' ≥ 1 minus the set of all primes) or a large set (e.g. the set of all positive integers minus the set of all positive even numbers). In set theoretic terminology, the small sets form an [[ideal (set theory)|ideal]].
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| * The [[Müntz–Szász theorem]] is that a set <math>S=\{s_1,s_2,s_3,\dots\}</math> is large if and only if the set spanned by
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| :<math>\{1,x^{s_1},x^{s_2},x^{s_3},\dots\} \,</math>
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| : is [[dense set|dense]] in the [[uniform norm]] topology of [[continuous function]]s on a closed interval. This is a generalization of the [[Stone–Weierstrass theorem]].
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| ==Open problems==
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| There are many sets about which it is not known whether they are large or small.
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| Not known how to identify a large set or a small set, except proving by exhaustion.
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| [[Paul Erdős]] famously asked the [[Erdős conjecture on arithmetic progressions|question]] of whether any set that does not contain arbitrarily long [[arithmetic progression]]s must necessarily be small. He offered a prize of $3000 for the solution to this problem, more than for any of his [[Erdős conjectures|other conjectures]], and joked that this prize offer violated the minimum wage law.<ref name="pomerance">[[Carl Pomerance]], [http://www.ams.org/notices/199801/vertesi.pdf Paul Erdős, Number Theorist Extraordinaire]. (Part of the article ''The Mathematics of Paul Erdős''), in ''[[Notices of the AMS]]'', [http://www.ams.org/notices/199801/index.html January, 1998].</ref> This question is still open.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * A. D. Wadhwa (1975). An interesting subseries of the harmonic series. ''American Mathematical Monthly'' '''82''' (9) 931–933. {{JSTOR|2318503}}
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| [[Category:Combinatorics]]
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| [[Category:Integer sequences]]
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| [[Category:Mathematical series]]
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