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254:. Even density is problematic since many interesting sequences are of zero density despite having different intuitive sizes. The usual combinatorial definition of a large set is one with a divergent reciprocal sum, with the corresponding definition of a small set as one with a convergent reciprocal sum. Thus the natural numbers are large (since the
40:. Please do not link to this page from the article namespace. When finished, the page may be incorporated into another article or used as the basis for a new article. While in this unfinished state, the text is dual-licensed under the GFDL and CC-BY-SA. If you would like me to release it to the public domain, please
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is a generalization that looks for the size of the powers in this measure. The cubes are thus smaller than the squares in this sense, since their degree is larger (9). An example of a small set in this sense is the powers of 2. (See also
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to be chosen if a random natural number is selected. Because the density may not exist in general, the upper density is often used instead. Instead of the limit, the
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is a sequence for which all sufficiently large numbers can be expressed as the sum of some subsequence. Essentially it is an additive basis of order ω.
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of the natural numbers has at least one part with arbitrarily-long arithmetic sequences with differences in the set. The natural numbers are large by
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a probabilistic definition is useful. Consider the chance of choosing an element of the set out of the first
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restricts the number of different cardinalities, if used. Large sets are usually taken to mean
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117:(with respect to some universe, usually the natural numbers) is finite. Similarly, in
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183:. Large sets have positive density, while small sets have density zero. Thus the
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is called its degree. Alternately, a set is an additive basis if a finite
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means that all large sets have arbitrarily long arithmetic progressions.
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means that there are an infinite number of different cardinalities. The
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Neil
Hindman and Dona Strauss, "Density in arbitrary semigroups",
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there are different formalizations of the intuitive concepts of
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natural numbers, then take the limit of this value as
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Series A, Vol. 113, No. 7 (Oct 2006), pp. 1219–1242.
300:such that every natural number is a sum of at most
612:Multiplicative structures in additively large sets
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566:E. Szemerédy, "On sets of integers containing no
318:is that the squares are large with degree four.
622:Ergodic Theory and the Properties of Large Sets
187:numbers are large (with density 0.5) while the
586:S. A. Burr, P. Erdos, R. L. Graham and W. Li,
610:Beiglböck, Bergelson, Hindman, and Strauss. "
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588:Complete sequences of sets of integer powers
296:, that is, if there exists a finite number
274:Erdős conjecture on arithmetic progressions
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258:diverges), the prime numbers are large (
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312:of the set equals the natural numbers.
570:-elements in arithmetic progression",
304:elements from the set. The least such
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270:, the set of twin primes is small.
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149:as well as many other fields like
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131:are another type of 'small' set.
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125:sets could be considered large.
92:generalized continuum hypothesis
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616:Journal of Combinatorial Theory
494:, a set is large if any finite
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316:Lagrange's four-square theorem
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1:
292:, a set is large if it is an
191:are small (density 0 by the
176:{\displaystyle n\to \infty }
113:sets: a set is large if its
390:Small set (category theory)
247:{\displaystyle \aleph _{0}}
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500:Van der Waerden's theorem
486:Large set (Ramsey theory)
219:Small set (combinatorics)
195:). Sets of density 1 are
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516:piecewise syndetic set
440:prevalent and shy sets
290:additive number theory
284:Additive number theory
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109:Alternately, consider
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597:(1996), pp. 133-138.
577:(1975), pp. 199–245.
557:(2006), pp. 273–300.
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193:prime number theorem
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50:In many branches of
207:Szemerédi's theorem
38:preliminary version
459:. You can help by
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363:. You can help by
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605:Further expansion
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332:complete sequence
278:Green–Tao theorem
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197:almost sure
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106:are small.
104:finite sets
84:cardinality
68:bornologies
52:mathematics
538:References
436:Meagre set
203:is taken.
129:Porous set
121:universes
115:complement
80:set theory
74:Set theory
64:set ideals
42:contact me
520:thick set
496:partition
372:July 2012
236:ℵ
171:∞
168:→
100:countable
526:See also
512:syndetic
468:May 2010
430:Topology
418:May 2010
111:cofinite
266:). By
201:lim sup
506:Others
325:IP set
310:sumset
189:primes
66:(even
260:proof
60:small
56:large
16:<
438:and
185:even
102:and
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