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'''Dyson's transform''' is a fundamental technique in [[additive number theory]].<ref name=nathanson>Additive Number Theory: Inverse Problems and the Geometry of Sumsets
By Melvyn Bernard Nathanson, Springer, Aug 22, 1996, ISBN 0-387-94655-1, http://books.google.com/books?id=PqlQjNhjkKUC&dq=%22e-transform%22&source=gbs_navlinks_s, p. 42</ref> It was developed by [[Freeman Dyson]] as part of his proof of [[Mann's theorem]]<ref name="Sequences">
{{cite book | last1 = Halberstam | first1 = H. |authorlink1 = Heini Halberstam | last2 = Roth | first2 = K. F. | authorlink2 =Klaus Roth | title = Sequences | publisher = Springer-Verlag | location = Berlin | year = 1983 | edition = revised | isbn = 978-0-387-90801-4 }}</ref>{{Rp|17}}, is used to prove such fundamental results of Additive Number Theory as the [[Cauchy-Davenport theorem]],<ref name=nathanson /> and was used by [[Olivier Ramaré]] in his work on the [[Goldbach conjecture]] that proved that every even integer is the sum of at most 6 primes.<ref name="Ramaré">
{{cite journal | author=O. Ramaré | authorlink=Olivier Ramaré | title=On šnirel'man's constant | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV | volume=22 | year=1995 | issue=4 | pages=645–706 | url = http://www.numdam.org/item?id=ASNSP_1995_4_22_4_645_0 | accessdate = 2009-03-13}}
</ref>{{Rp|700–701}} The term ''Dyson's transform'' for this technique is used by Ramaré.<ref name="Ramaré" />{{Rp|700–701}} Halberstam and Roth call it the τ-transformation.<ref name="Sequences"/>{{Rp|58}}


This formulation of the transform is from Ramaré.<ref name="Ramaré" />{{Rp|700–701}} Let ''A'' be a sequence of natural numbers, and ''x'' be any real number. Write ''A''(''x'') for the number of elements of ''A'' which lie in [1,&nbsp;''x'']. Suppose <math>A= \{a_1<a_2< \cdots\}</math> and <math>B= \{0=b_1<b_2<\cdots\} </math> are two sequences of natural numbers. We write ''A''&nbsp;+&nbsp;''B'' for the [[sumset]], that is, the set of all elements ''a''&nbsp;+&nbsp;''b'' where ''a'' is in ''A'' and ''b'' is in B; and similarly ''A''&nbsp;&minus;&nbsp;''B'' for the set of differences ''a''&nbsp;&minus;&nbsp;''b''. For any element ''e'' in ''A'', Dyson's transform consists in forming the sequences <math> A'= A \cup \{B + \{e\}\}</math>  and  <math>\,B'= B \cap \{A - \{e\}\}</math>. The transformed sequences have the properties:


* <math>A' + B' \subset A + B </math>
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* <math>\{e\} + B' \subset A' </math>
* <math>0 \in B' </math>
* <math>A'(m)+ B'(m-e) = A(m) + B(m-e) \ </math>
 
==References==
{{Reflist}}
 
[[Category:Sumsets]]
[[Category:Freeman Dyson]]
 
 
{{numtheory-stub}}

Latest revision as of 14:14, 7 May 2014


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